Associative polymers for mist-control

ABSTRACT

Polymeric mist control materials, methods of forming polymeric mist control materials, and methods of using such materials for mist control are provided. The polymeric mist control additives are formed of molecules comprised predominantly of monomers that confer high solubility in fuel and include associative groups that attract each other in donor-acceptor manner, and are incorporated such that multiple associative groups are in close proximity (“clusters”), such that the clusters are separated by very long non-associative sequences.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/763,144, filed Apr. 19, 2010, which claims priority to U.S.Provisional Application No. 61/170,271, filed Apr. 17, 2009. Thedisclosure of each of these applications is incorporated herein byreference in its entirety.

FIELD OF THE INVENTION

This invention relates generally to polymeric additives to suppressmisting, more particularly to polymeric fuel additives that confer highsolubility in fuel and include associative groups that attract eachother in donor-acceptor manner, to methods of making such materials, andmethods of suppressing misting using such materials.

BACKGROUND OF THE INVENTION

The problem of fuel fires is ubiquitous for vehicles ranging fromautomobiles to jumbo jets and for fuel handling operations, particularlydispensing. In the field of civil aviation, over the last four decades,on average two air carrier accidents have occurred monthly within the USand fifteen worldwide. This level persists despite ongoing efforts toeliminate human error and improve security. Of these, an estimated 70%occur on takeoff or landing and are impact-survivable. It is furtherestimated that 40% of the fatalities in such crashes are due to firecaused by combustion of aviation fuel. Thus, it is estimated that some500 to 1000 lives per year can be saved by the development of aneffective mist control fuel system.

Perhaps more important is the issue of homeland security. Thedestructive power of a fully-fueled aircraft comes from the fuel—notkinetic energy. For example, in the case of the Sep. 11, 2001 attack onthe World Trade Centers, both towers absorbed the aircraft's momentumand survived the initial impact. Threats to high-rise buildings, sportsarenas, nuclear power facilities, and other important targets resultfrom the explosion and intense post-crash fire. Thus, the successfulincorporation of a mist control fuel additive would greatly reduce theproperty loss caused by plane accidents, and may serve to deterterrorists from using passenger aircraft fuel as a weapon of massdestruction.

Presently, there are no implemented technologies to reduce the firehazard of fuel in crash scenarios. The associated safety and securityissues are merely accepted as risks incident to air transportation.Attempts in the past to mitigate aircraft crash fires have led to theincorporation of firewalls, flame arresters, fuel-line isolation, fireextinguishing systems, fire-resistant materials, etc. with limitedsuccess. A 1997 NRC report points to this problem specifically, stating,“the reduction of the fire hazard of [the] fuel [itself] is critical inimproving survivability in past crashes.” (See, e.g., NCR Proceedings,NMAB-490, Washington D.C., 1997, the disclosure of which is incorporatedherein by reference.) Candidate technologies aimed to provide apost-impact fire-safe fuel were evaluated in the January 2000 report ofSouthwest Research Institute, prepared by Bernard Wright under contractto NASA. (See, e.g., “Assessment of Concepts and Research for CommercialAviation Fire-Safe Fuel,” Bernard Wright, SWRI, January 2000, thedisclosure of which is incorporated herein by reference.) Based on anextensive review of fuel vulnerability studies and discussions withindustry-knowledgeable sources, Mr. Wright concluded thatMist-Controlled Kerosene (MCK) technology was rated as the mostpromising and highest priority to date.

MCK is a conventional Jet-A fuel to which a small fraction of high MWpolymer (<0.3 wt %) has been added. When fuel is released from rupturedtanks into the airflow around a crashing aircraft, the polymerinterferes mechanically with the formation of mist. Linear polymerchains that have high enough MWs are well known to be effectivemist-control agents (i.e. of MW>˜5,000,000 g/mol, abbreviated as 5Mg/mol). Unfortunately, ultralong homopolymers degrade upon passagethrough pumps, filters or long pipelines (termed “shear degradation”),rendering them ineffective for mist suppression. Attempts to createpolymers that suppress misting and resist flow degradation have beenmade, primarily by adding randomly placed associating groups ontofuel-soluble polymers. Among the large number of associating polymerssynthesized in an attempt to achieve mist-control the one that advancedthe farthest technologically was “FM-9”, a proprietary polymer developedby the British company Imperial Chemical Industries (ICI) in the 1970's.The last major attempt at polymer-modified fuel was the FAA-fundedanti-misting kerosene (AMK) program centered on FM-9, shown in FIG. 1 a.By trial and error a fuel formulation was produced that reduced thepost-crash fire; however, ultimately the program failed because theformulation could not be produced, transported and stored using existingfuel handling systems. It did not have adequate resistance to sheardegradation, so it could not be pumped or filtered (therefore, theadditive could not be incorporated at the point of production of thefuel or in any centralized facility). Efforts to add the polymers to thefuel at the point of delivery have proven impractical both economicallyand from a regulatory point of view. Furthermore, FM-9 tended to phaseseparate, particularly under cold conditions, so it could not be storedor used at low temperature (e.g., it would deposit on the walls ofstorage tanks and would clog fuel filters). The FM-9 polymers lostefficacy under hot conditions due to disruption of the attractionbetween associating groups.

Several groups have found experimentally that by placing associativegroups randomly along polymer chains, the molecules could be made to‘stick’ to each other in aggregates of approximately 10 to 100 chains.Therefore, aggregates greater than 5M g/mol can be achieved usingpolymers of MW 0.1M-1M g/mol that are small enough that they do notundergo shear degradation during fuel handling. Some experimentsindicated that flow degradation could be reduced using randomly placedassociative groups (because aggregates could pull apart during briefintervals of intense flow and reassemble again afterward.) Nevertheless,despite the extensive effort devoted to randomly functionalizedassociative polymers, no commercially viable mist control fuel has yetbeen produced. The polymers were not effective at the dilute conditionsof most interest for fuel additives (for example, FM-9 was used at 0.3%wt, a much greater concentration than any other fuel additive). Atdilute concentrations, the associating groups within a given polymerdrive the polymer to collapse upon itself, rendering it ineffective formist suppression. Furthermore, addition of associating groups to afuel-soluble polymers tends to drastically reduce its solubility infuel, leading to phase separation, which leads to unacceptable behaviorduring storage and use (noted above in relation to FM-9)

In an attempt to remedy the loss of multi-chain clusters at diluteconcentrations, a few studies have examined “donor-acceptor” associativepolymers that use two different polymers, one bearing randomly placed“donor” groups (that do not associate with each other) and the otherpolymer bearing randomly placed “acceptor” groups (that do not associatewith each other). The driving force for association causes “donorchains” to associate with “acceptor chains”, even under diluteconditions. Unfortunately, as will be fully developed below, all ofthese prior polymer design concepts aimed at reducing the fire hazard offuel were misguided. Even supramolecules held together by association ofrandomly distributed donor and acceptor groups exhibit chain collapseunder dilute conditions: the multi-chain aggregates are densely “stuck”to one another and occupy a much smaller volume than theunfunctionalized, separate chains would. Despite the high molar mass ofthe aggregate, they are less effective for mist suppression than thecorresponding homopolymers (with no associative groups at all).

Accordingly, despite decades of effort, no polymer design has beendiscovered that can overcome shear degradation and avoid chain collapse,and thereby provide effective mist control. The current inventiondescribes mist control polymers that have the following properties:

-   Can be added at the refinery where other fuel additives e.g.,    anti-static agents, are introduced;-   Provides effective fire protection at low concentrations between    50-500 ppm;-   Withstands unintentional degradation during fuel handling;-   Does not deposit onto materials used in storage tanks, filters and    transfer systems;-   Will be compatible with current aviation fuel handling and pumping    systems; and-   May be synthesized at an acceptable cost.

BRIEF SUMMARY OF THE INVENTION

Thus, there is provided in accordance with the current invention apolymeric associative mist control material including a plurality ofmedium-soluble chains and a plurality of associative groups, wherein theassociative groups attach in clusters at the ends of the medium-solublechains such that these clusters of associative groups are separated bythe medium-soluble chains.

In one embodiment, the polymers of the invention are telechelic andinclude a mixture of at least two different complementary polymers. Insuch an embodiment, each of the polymers includes one or more long,non-associative middle blocks selected to confer solubility to thepolymer in a particular medium, and a plurality of associative groupsdisposed in small clusters that are separated by the long, fuel-solubleblocks. The different complementary polymers reversibly associate withan association strength less than that of a covalent bond to formsupramolecules of sufficient size to prevent misting of the medium.

In another embodiment, each of the clusters comprises at least twentyassociating groups. The associative groups are placed in clusters sothat their overall strength of association can be increased to thestrength required to remain effective to the highest temperaturedemanded by type of fuel and its use conditions. In another suchembodiment, the clusters are formed using a mode selected from the groupconsisting of comonomers, dendrimers, nanoparticies and speciallydesigned chemical units that confer polyvalent association. In all suchembodiments, the associative groups on a particular polymer do notassociate with themselves.

In still another embodiment of the invention, the complementary polymershave a molecular weight of from 100 to 1000 kg/mol.

In yet another embodiment of the invention, the middle chains conferhigh solubility in fuel over a wide temperature range −40° C. to +60° C.In such an embodiment, the solubility in fuel is conferred to the middlechains by hydrocarbon groups, such as alkanes and alkenes, in thepolymer backbone and attached thereto.

In still yet another embodiment of the invention, the differentcomplementary polymers form a donor/acceptor pair.

In still yet another embodiment, the associative groups are selectedfrom the group of hydrogen bond donor/acceptor pairs, charge transferdonor/acceptor pairs and ionic bond donor/acceptor pairs. In one suchembodiment, the associative groups include a carboxylic acid/tertiaryamine acid/base pair. In such an embodiment, the tertiary amine sidegroups may be selected from the group consisting of dialkylamino groupsconnected via a linker that includes methylene carbons adjacent to theamine, such as (dimethylamino) alkyl groups, (diethylamino) alkylgroups, (methylethylamino) alkyl groups, pyridyl groups, etc. In anothersuch embodiment, the associative groups include a hydrogen bond pairselected from the group consisting of hydroxyphenyl/pyridyl pairs and2,4-Bis(propylamido)pyridine/Hydrouracil pair. In another suchembodiment, the associative groups include a charge transfer pair. Insuch an embodiment, the electron donor is selected from the groupconsisting of carbazole derivatives and N,N-dimethylanilino derivatives,and the electron acceptor is selected from the group consisting ofdinitrophenyl group; 3,6-dinitro-carbazolyl group; buckminsterfullerene(C₆₀); 4-bromo-1-naphthyl group, dansyl group, anthracene, pyrene,2,3,5,6-tetrafluorophenyl group; and cyanophenyl group.

In still yet another embodiment, the fuel-soluble blocks arehomopolymers or copolymers of different monomers. In one suchembodiment, the fuel-soluble block is an unsaturated hydrocarbon. Insuch an embodiment, the monomers are selected from the group ofisoprene, butadiene, ethylene, propylene, butene, norbornenederivatives, cyclobutene, cyclopentene, cyclooctene, cyclooctadiene, andtrans,trans,cis-1,5,9-cyclododecatriene.

In still yet another embodiment, the association of the complementarypolymers is sufficiently strong to provide self-assembly of the polymersinto supramolecules at temperatures up to 60° C.

In still yet another embodiment, the mist control material comprises amixture of two complementary polymers, where the associative groups ofeach of the complementary polymers is different such that each end ofeach of the complementary polymers is designed to associate with onlyone end of the other complementary polymer. In one such embodiment, theassociative groups at the first end of both complementary polymersassociate by an acid/base interaction, and the associative groups at thesecond end of both complementary polymers associate by an electrondonor/acceptor interaction. In one preferred embodiment, thesupramolecules are linear chains comprised of a defined number ofcomplementary polymers. In one such embodiment, the complementarypolymers controllably associate to form linear pentamer supramolecules.

In still yet another embodiment, the polymers are provided in aconcentration of from 100 to 1000 ppm.

In still yet another embodiment, the invention is directed to a methodof forming mist control materials, comprising synthesizing at least twodifferent complementary telechelic polymers, combining the complementarypolymers to form a mixture, and allowing the associative groups of theat least two different complementary polymers to reversibly associatewith an association strength less than that of a covalent bond to formsupramolecules of sufficient length to prevent misting of the medium. Inone preferred embodiment, the supramolecules are linear moleculescomprised of a defined number of complementary polymers.

In still yet another embodiment, the associative groups are formed bycontrolled radical polymerization. In another such embodiment, thecontrolled radical polymerization technique is chosen from the groupconsisting of atom-transfer radical polymerization and reversibleaddition-fragmentation chain transfer polymerization. In another suchembodiment, the telechelic polymers are synthesized byruthenium-catalyzed ring-open metathesis polymerization. In one suchembodiment, the synthesis of the telechelic polymers is conducted in thepresence of a custom chain transfer agent.

In still yet another embodiment, the mist control material is a mixtureof associative polymers formed by condensation of medium-soluble chainsand clusters of associative groups.

In still yet another embodiment, the invention is directed to a methodof controlling mist in a medium comprising providing a mixture ofassociative polymers wherein the associative groups of the at least twodifferent complementary polymers may reversibly associate with anassociation strength less than that of a covalent bond to formsupramolecules designed to prevent misting of the medium atconcentrations of from 50 to 500 ppm. In one such embodiment theassociative polymers comprise a mixture of at least two differentcomplementary telechelic polymers.

BRIEF DESCRIPTION OF THE DRAWINGS

The description will be more fully understood with reference to thefollowing figures and data graphs, which are presented as exemplaryembodiments of the invention and should not be construed as a completerecitation of the scope of the invention, wherein:

FIG. 1 provides schematics of conventional polymer chains bearingrandomly distributed functional groups, including: (a) Self-associatingFM-9 random copolymer, (b) self-associating triazolinedione-modified 1,2PB, and (c) hydrogen bond donor/acceptor copolymer pair synthesized from1,2 PB;

FIG. 2 provides a general scheme for the synthesis of mist controlsupramolecules from polymer chains in accordance with one embodiment ofthe invention;

FIG. 3 provides a general scheme for the self-assembly of mist controlagents in accordance with one embodiment of the invention in which loopsare prohibited and associations result in the formation of well-definedlinear chains;

FIG. 4 provides a scheme for the self-assembly of mist control agents inaccordance with one embodiment of the instant invention comprisingtelechelic chains with associative clusters at each end, where (a) showsthe synthesis of precursor short chains that have double bonds in themiddle and an initiator “R” at each end (“X” denotes an alkane (e.g.,cyclopentane or hexane, (b) shows the synthesis of a precursor of amacro-chain transfer agent through ROMP, (c) shows the growth ofassociating blocks using CRP, and (d) representative cycloolefinmonomers suitable for ring opening metathesis polymerization that can beused individually or in combination to produce long, fuel-solublehomopolymer or copolymer chains in the presence of the desired chaintransfer agent, CTA;

FIG. 5 provides a scheme for the synthesis of an exemplary mist controlmaterial in accordance with the current invention, specifically theembodiment illustrated in FIG. 3, where (a) shows an ABA triblockprepared by attaching hydrogen bond acceptors to vinyl groups clusteredin each of the end-blocks of the prepolymer, (b) shows an AB diblockprepared by attaching charge transfer electron acceptors to vinyl groupsclustered at one end of the prepolymer, and (c) shows an ABC triblockwith hydrogen bond donors at one end and charge transfer electron donorsat the other end, prepared by attaching charge transfer groups to vinylgroups that were present only in one end-block of the prepolymer;

FIG. 6 provides molecular designs for self-assembly of polymericbuilding blocks into larger supramolecules via physical interactions fora pair of telechelic polymers, one with donors clustered at both of itsend and the other with acceptors clustered at both of its ends, leadingto a series of equilibriums between linear supramolecules and cyclicsupramolecules;

FIG. 7 provides a schematic of the two different telechelic polymercomponents, and the supramolecules that they form;

FIG. 8 provides data plots for model predictions for strength ofinteraction εkT=14 kT (left) and εkT=16 kT (right);

FIG. 9 provides data plots for model predictions for strength ofinteraction εkT=18 kT (left) and εkT=20 kT (right);

FIG. 10 provides a schematic of exemplary end caps for the polymercomponents;

FIG. 11 provides data plots for model predictions in the presence ofend-capped chains, when εkT=16 kT;

FIG. 12 provides data plots for model predictions in the presence ofend-capped chains, when εkT=18 kT;

FIG. 13 provides data plots for model predictions in the presence ofend-capped chains, when εkT=20 kT;

FIG. 14 provides data plots for the limiting equilibrium distribution(as ε→∞) obtained in the presence of end-capping A---- polymer chains;

FIG. 15 provides a molecular diagram of sextuple hydrogen-bonding motifsderived from nucleobase structures, of binding constants˜10⁶ M⁻¹ inorganic solvents of low polarity; and

FIG. 16 provides a data plot of the relationship between shear viscosityand shear rate in exemplary mist control materials.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to associative polymeric additivesthat can be used for mist control, methods of forming such mist controlmaterials, and methods of suppressing misting using such materials. Thepolymeric mist control additives in accordance with the currentinvention are formed of complementary polymers comprised predominantlyof monomers that confer high solubility in a medium of choice, such as,for example, fuel, and include associative groups that attract eachother in donor-acceptor manner, and are incorporated such that multipleassociative groups are in close proximity (“clusters”) such that theclusters are separated by one or more nonassociative sequences.Exemplary modes of incorporating clusters of associative groups into thepolymers of the invention include comonomers, dendrimers, nanoparticiesand specially designed chemical units that confer polyvalentassociation. In the invention, donor˜acceptor interactions may be chosenfrom a variety of complementary pairs, including, but not limited to,hydrogen bond donor/acceptor pairs, charge transfer donor/acceptor pairsand ionic bond donor/acceptor pairs.

The polymer structure of the invention fulfills essential requirementsfor mist control that are unprecedented, including, excellent sheartolerance, efficacy at very low concentration, solubility even at a userspecified lower use temperature, the ability to tune associationstrength so that efficacy is retained to a user specified upper usetemperature, etc. As a result, the current invention provides a novelclass of polymeric mist control materials.

Definitions

Block: for the purposes of this invention, is used in the same sense asin the literature on block copolymers, meaning a consecutive sequence ofrepeat units of a given type (such as, fuel-soluble repeat units). (See,e.g., Ergungor, Z., Journal of Non-Newtonian Fluid Mechanics, 97 (2-3),Pp. 159-167 (2001), the disclosure of which is incorporated herein byreference.)

Long Block: for the purposes of this invention, refers to a sequence of100 or more repeat units.

Cluster: for the purposes of this invention, is a collection of multipleassociative groups in close proximity. Such clusters of associativegroups may be incorporated into the polymers of the invention via anumber of methods, including, for example, comonomers, dendrimers,nanoparticies and specially designed chemical units that conferpolyvalent association.

Comparison with Conventional Mist Control

Mist-controlled kerosene (MCK), previously documented as anti-mistingkerosene (AMK), is a conventional hydrocarbon fuel to which a smallfraction of high molecular weight polymer has been added. (See, e.g., M.Yaffee, DOT/FAA/CT-86/7, April 1986, the disclosure of which isincorporated herein by reference.) When fuel is accidentally spilled,the polymer interferes mechanically with the formation of mist. Longchain molecules hold the fuel together in much larger drops and, thus,act as a mist-control agent. The total surface area available tovaporize is several orders of magnitude less for the mist-controlledfuel's large drops than the fine mist generated by the conventionalfuel. This vast reduction in vapor and the increase in the thermal massper drop combine to reduce the rate of energy release in the presence oftransient ignition sources during tank rupture, which further reducesvaporization relative to untreated fuel. Any resulting fire is cooler,tends not to propagate away from the ignition source, self-quenches whenthe ignition source is removed, provides considerable cooling to exposedsurfaces, and the likelihood of igniting pool fuel fires is eithereliminated or significantly delayed. On the other hand, fine fuel mistburns uncontrolled if ignited and the resulting fire propagates awayfrom the ignition source involves more fuel, and thus triggering deadlypool fires. Fuel mist triggered pool fires are known to be very violentand intense, often accompanying tank explosions, leaving no chance tointervene. Accordingly, mist-controlled fuel eliminates or significantlydelays tank explosions, giving personnel time to move away from the siteand giving firefighters critical time to contain and extinguish thefire.

The obstacle to implementation of MCK over the past thirty years hasbeen the shear degradation of the high molecular weight polymers.Specifically, as the fuel is pumped during transportation anddispensing, the very long polymer shear degrades rapidly and loses itsmist-control capability. The current invention presents a novel familyof associative mist-control polymers, which overcome this long-standingshortcoming. However, prior to examining the associative mist controladditives of the current invention, it is important to understand thedeficiencies of past mist control polymers.

Past research has shown that the presence of ultra-high molecularpolymer in fuel, e.g., JP-8, diesel fuel, Jet-A etc. suppresses mistingand could provide the necessary fire suppression needed. Polyisobutylene(PIB) is an excellent example of such polymers additives. However, thereare several operational issues with such polymers as mist-controladditives (Table 1), such as unintentional shear degradation. Suchpolymer degradation, which will result in loss of fuel fire protectionin operation, has to be overcome before one could fully realize itsbenefits as a mist-control additive. The inventive polymers associateinto ultralong chains with non-covalent linkages that are weaker than acovalent bond and thereby, by design, overcome this limitation. Duringpassage through pumps, for example, the non-covalent linkagesdisassemble, preventing covalent bond rupture. The inventive chainsrapidly re-associate after passing through an intense flow to restorethe supramolecular structures that provide mist control. While someassociative polymers—including FM-9—have been examined over the past 30years, no polymer has successfully met all of the criteria satisfied bythe materials of the instant invention. Most of the prior efforts weredevoted to polymers in which the distribution of associative groupsalong the chain was random. The extensive investigation leading to theinstant invention showed that the use of randomly-placed associativegroups causes the chains to collapse. That is the molecular-level reasonthat relatively high concentrations (>3000 ppm) of FM-9 were needed inorder to be effective in mist suppression. (See, e.g., Knight, J. U.S.Pat. No. 4,396,398, the disclosure of which is incorporated herein byreference.) In the dilute regime of interest for fuel additives, apolymer with randomly-placed associative groups loses its efficacy andis even less effective than linear non-associating chains of the samemolecular weight. (See, e.g., David, R. L. A., Ph.D. Dissertation,California Institute of Technology, Pasadena, Calif., 2008, thedisclosure of which is incorporated herein by reference.) In addition,the research leading to the instant invention showed thatrandomly-placed associating sites lead to phase separation. That is thereason that FM-9 tended to phase-separate, which precluded storage offuel containing the additive and caused problems when the fuel was usedat low temperatures. In addition to chain collapse, FM-9 also sufferedfrom loss of efficacy at high temperature and high affinity to water(Table 1). The research leading to the instant invention further showedthat inherent limitations of polymers with randomly-placed associativegroups are not remedied by using donor˜acceptor interactions betweenrandomly-distributed associative groups. Thus, there were no successfuldesign principles for mist-control polymers, despite decades of prioreffort. The associative polymers of the instant invention prevent sheardegradation and chain collapse; remain in solution at low temperature;and remain effective at high temperature. The resulting advantages ofthe inventive polymers over past mist-control polymers and the currentstate of the associative polymer research are summarized in Table 1,below.

TABLE 1 Comparison of Conventional and Inventive Polymers PriorAssociative Criteria PIB > 10M Polymers Caltech Polymers Logistics (−)must be added at (+) can be added (+) added at the time of fueling,prior to fueling, refinery-stable with because it is because it resistsrespect to pumping, destroyed during shear degradation. temperature,filtration pumping (−) phase separates, clogging filters Efficacy (−)only works until it (−) less effective than (+) as effective as >1Mpasses through pumps the non-associating linear polymer polymer would be(+) keeps efficacy even when pumped Combustion (+) polymer burns (+)very few (+) polymer burns cleanly heteroatoms (O, N), cleanly (>99.9%C, H) within the specifications of fuel Field (+) remains soluble (−)phase separates at (+) does not phase Temperature over entire field Tlow T (clogging separate at low T Range range (−40° F. to filters andsettling in (+) remains active at high 140° F.) fuel tanks) T (−)associations fail at high T

Discussion of Inventive Associative Mist Control Polymers

As discussed above, previous attempts to form mist control polymers havefocused on linear chains possessing associating functional groupsgrafted at random positions along the entire chains. What thesetechniques fail to appreciate is that randomly placed ‘stickers’ causechains to collapse. Specifically, the scientific fundamentals of self-and complementary-association of polymer chains (see, e.g., David, R. L.A., “Associative polymers as anti-misting agents and other functionalmaterials via thiol-ene coupling”, Ph.D. Dissertation, CaliforniaInstitute of Technology, Pasadena, Calif., 2008; David, R L A; et al.,Macromolecules, 42, 1380-1391 (2009); and David, R L A; et al., Polymer,50, 6323-6330 (2009),), the disclosures of each of which areincorporated herein by reference) show that the chain-collapse andsolubility issues of prior associative polymers for mist control, suchas, for example, FM-9 arise from random placement of associating groupsalong polymer backbones.

The current invention describes associative mist control polymers thatare controllably associated. In the development of the associative mistcontrol polymers of the current invention, molecular designs involvingboth self-associating interactions and donor-acceptor interactions werestudied, and the effects of extent of functionalization as well asconcentration of polymer components on shear and extensional rheology ofdilute solutions in non-polar hydrocarbon solvents were examined. Aswill be discussed in greater detail in the exemplary embodiments below,these studies have made it clear that not all associative polymers canbe effective for mist control. For example, it has been discovered thatfor both self-associating and donor-acceptor systems, intra- andintermolecular associations cause collapse of chains with randomlydistributed associative groups and interfere with the mechanism of mistcontrol by inhibiting stretching of the chain in extentional flow. Basedon these observations, it has been discovered that clusteringassociative groups and connecting them via long, fuel-soluble chainsmitigates chain collapse, particularly when complementary pairs ofassociating groups are used.

Accordingly, the current invention provides a molecular design ofassociative mist-control polymer additives comprising groupings orclusters of associative groups separated by one or more long blocks thatrender the mist control additive soluble in a particular medium, suchas, for example, aviation fuel. For the proper functioning of the mistcontrol materials, the associative groups of the instant invention musthave two key properties: they must not self-associate and they mustassociate with a strength that is less than that of a covalent bond.

In one preferred embodiment, shown schematically in FIG. 2, the mistcontrol additives of the instant invention are binary mixtures of donorand acceptor polymers or chains with complementary associating groupsclustered near the chain ends (i.e., “telechelic” polymers, FIG. 2,left). In the mist control additives of the current invention, themajority of the chain is a polymer that provides solubility in fuels,whereas clusters of associative groups, such as short blocks withassociative side groups, provide reversible association holding polymerchains together into very long “super chains” (FIG. 2, right). In apreferred embodiment, the donor chains are triblock copolymers with avery long fuel-soluble midblock and two short endblocks that preferablycontain more than five associative donor groups, and the acceptor chainsare triblock copolymers with a very long fuel-soluble midblock and twoshort endblocks that preferably contain more than five associativeacceptor groups.

Although many complementary polymers may meet the broad structuralrequirements discussed above, there are two fundamental requirements ofthe mist control additives of the instant invention that must be met.Specifically, the mist control materials must be made of individualpolymer chains that are short enough to survive flow conditionsencountered in pumping fuel (FIG. 2, left); and yet are able to assembleand reassemble into supramolecules that are long enough to suppressmisting (FIG. 2, right). More specifically, the inventive complementarypolymers of the instant invention have the following characteristics:

-   Have specially designed length (between 100 to 1000 kg/mol) so they    do not degrade upon passage through pumps,-   Are formed of specially selected monomers that provide solubility to    the complementary polymers in a particular medium (such as, for    example fuel) over a wide temperature range (−40° C. to +60° C.)    such that the medium/mist control material mixture remains a    homogeneous, single phase during storage, and-   Have specially selected associative clusters that are designed to    have other important properties including:    -   the clusters do not self-associate (i.e., in FIG. 2, open circle        ends do not attract each other; likewise for the solid square        ends),    -   the clusters associate in donor-acceptor, complementary pairs to        form large supramolecular structures that are open and flexible        so they suppress misting,    -   the clusters form associations that are weaker than covalent        bonds (so they protect against shear degradation, allowing        supramolecular chains to pull apart without chemical degradation        during passage through pumps and filters), and    -   the clusters form associations that are strong enough to retain        efficacy at temperatures up to 40° C., and preferably 60° C.-   Although not essential, in a preferred embodiment, the complementary    polymers of the instant invention are also selected, such that the    supramolecules formed through their association are primarily of a    linear structure (i.e., the concentration of linear supramolecules    is greater than that of branched or cyclic supramolecules. In still    another embodiment of the invention, the supramolecules have a    well-defined length (i.e., a specific number of complementary    polymers combine together into the supramolecule).

Although it should be understood that this is not intended to be anexhaustive listing of applicable mist control materials that meet therequirements listed above, some exemplary polymers are provided here. Asa starting point, any polymer backbone that provides good solubility ina desired medium (e.g., fuels) and elasticity for mist-control may beused. In a preferred embodiment the fuel soluble domains used in thecurrent invention have a molar mass greater than 100,000 g/mol or adegree of polymerization greater than 1000. In a more preferredembodiment, the fuel-soluble domain has an unsaturated hydrocarbonstructure. In still another ROMP monomers (either homopolymers orcopolymers) with sufficient ring strain are chosen. Exemplaryembodiments of fuel-soluble domains include homopolymers or copolymerssynthesized by ring opening metathesis (ROMP) of one or more cyclicalkenes, including, but not limited to, Norbornene derivatives (forexample, norbornene, 5-Ethylidene-2-norbornene,alkyl-5-norbornene-2,3-dicarboximide (e.g.,N-(2-Ethylhexyl)-5-norbornene-2,3-dicarboximide), cyclooctene,cyclooctadiene, trans,trans,cis-1,5,9-cyclododecatriene, etc. Stillother exemplary embodiments of fuel-soluble domains includepolyisoprene, polybutadiene and their copolymers and derivatives thatcan be prepared by anionic or radical polymerization. Certain saturatedhydrocarbon polymers can serve as fuel-soluble domains, notablypolyisobutylene (PIB).

Likewise, the donor˜acceptor interactions may be chosen from a varietyof complementary pairs, including, but not limited to, hydrogen bonddonor/acceptor pairs, charge transfer donor/acceptor pairs and ionicbond donor/acceptor pairs. Although not meant to be exhaustive,exemplary associating groups for each category are listed as follows:

-   Acid/base pairs: carboxylic acid (proton donor)/tertiary amine    (proton acceptor), wherein the tertiary amine may include, for    example, (dimethylamino) alkyl groups, (diethylamino) alkyl groups,    (methylethylamino) alkyl groups, and pyridyl groups.-   Hydrogen bonding pairs: hydroxyphenyl/pyridyl group and    2,4-Bis(propylamido)pyridine/Hydrouracil pairs.-   Charge transfer pairs: where exemplary electron donors are carbazole    derivatives (e.g., carbazolyl group, 3,6-diamino-carbazolyl group,    (carbazolylmethylene)-aniline group); and N,N-dimethylanilino group,    and where the exemplary electron acceptors are dinitrophenyl group;    3,6-dinitro-carbazolyl group; buckminsterfullerene (C₆₀);    4-bromo-1-naphthyl group, dansyl group, anthracene, pyrene,    2,3,5,6-tetrafluorophenyl group, and cyanophenyl group.

Exemplary modes of incorporating these clusters of associative groupsinto the polymers of the invention may include any compatible method,including, for example, comonomers, dendrimers, nanoparticies andspecially designed chemical units that confer polyvalent association.

Although thus far the discussion has focused on the mist controlmaterials themselves, the current invention is also directed to methodsof controllably forming such mist control materials. There are manysynthetic routes to such materials, including some suitable forindustrial production. In general terms, however, the scheme shown inFIG. 3 is preferably implemented, where the A+B interactions involveacid-base associations (e.g., between phenol and tertiary aminemoieties), and the C+D interactions involve electron donor-acceptorassociations (e.g., between carbazole and dinitrobenzene derivatives.)One key advantage of this synthetic route is that the equilibriumconstants of association of the A+B and C+D endblocks can beconveniently modified by simply changing the number of functionalmoieties featured on the endblocks. In addition, because this syntheticscheme involves two sets of specific interactions, where A endgroupsinteract only with B endgroups, and C endgroups likewise only with Dendgroups, loops may be prohibited such that associations result in thesystematic formation of well-defined linear chains. For example, themolecules may be designed such that for a stoichiometric blend of thebuilding blocks, and at high enough binding affinity of the A+B and C+Dassociations, nearly all the polymer chains should assemble intopentamers (in 4 bond-forming events only) even at arbitrarily lowpolymer volume fraction ϕ_(total). As a result, satisfactory mistsuppression could be achieved with <100 ppm of A----A, B----C, and D----chains of size MW=10⁶ g/mol.

One exemplary synthetic pathway that meets the above criteria, and thatwould be capable of producing polymers that meet the design criteria formist control agents in accordance with the current invention, is shownin FIG. 4. In this embodiment, mist control agents are synthesized usinga combination of ruthenium-catalyzed ring-open metathesis polymerization(ROMP) and controlled radical polymerization (CRP). More specifically,in this embodiment, clusters of associating groups would be preparedusing CRP techniques, such as atom-transfer radical polymerization(ATRP) and reversible addition-fragmentation chain transferpolymerization (RAFT), while difunctional macroinitiators would beprepared using ROMP (as shown in FIG. 4a ). In this embodiment, tofacilitate subsequent addition to the ends of the polymer chains, thedesired clusters of associating groups may, in turn, be grown from eachside of pre-polymer (as shown in FIG. 4b ). The desired telechelicpolymers are, in turn, synthesized by performing ROMP in the presence ofa custom chain transfer agent (FIG. 4c ). In a preferred embodiment,these custom chain transfer agents have alkene units in the middle,flanked by clusters of associative groups providing a bridge betweenROMP and CRP.

Although the above provides a relatively generic discussion of apossible synthetic pathway for producing the mist control agents of theinstant invention, it should be understood that there are several designcriteria that should be used in selecting the monomers or chains used inthe synthesis:

-   The polymers should be designed to ensure good solubility in a    selected medium, such as, for example, fuels, clean combustion and    hydrophobicity. In the case of fuels, one means of attaining this    goal is to select polymers that have a composition similar to the    fuel itself (i.e., polymers that incorporate alkane and alkene    units).-   The medium (e.g., fuel) soluble domains should be produced with high    MW and clusters of associating groups should be used to ensure high    conversion of the chains to supramolecules. For example, using the    above technique it is possible to produce telechelic chains with a    total molecular weight of roughly 5×10⁵ g/mole with approximately 20    associating groups at each end. Within these parameters, the    distribution of supramolecules will include a significant population    having size 5 million g/mol or more—sufficiently large for mist    control.-   The associative clusters should be selected and attached to the    chains such that they the following properties:    -   the clusters do not self-associate,    -   the clusters associate in donor-acceptor, complementary pairs to        form large supramolecular structures that are open and flexible        so they suppress misting,    -   the clusters form associations that are weaker than covalent        bonds (so they protect against shear degradation, allowing        supramolecular chains to pull apart without chemical degradation        during passage through pumps and filters), and    -   the clusters form associations that are strong enough that they        retain efficacy at temperatures up to 40° C., and preferably up        to 60° C.

One exemplary embodiment of specific polymeric materials that meet thatabove criteria and that can be used with the synthetic pathway describedin FIG. 4d , would be to use cyclic olefins with multiple carbon-carbondouble bonds, such as, for example, 1,5-cyclooctadiene (COD) andtrans,trans,cis-1,5,9-cyclododecatriene (CDT). These materials can giveexcellent solubility in fuels, but the lack of strong ring strain limitsthe molecular weights. However, copolymerization of these monomers withrelatively strained cyclic monomers (e.g., norbornene, 1-cyclooctene)that each having a single double can be used to achieve both solubilityand high molecular weight. In such an embodiment, complementary pairs ofassociating groups are chosen to give strong association in fuels. Inaddition, both hydrogen donor/acceptor pairs (e.g., carboxylicacids/tertiary amines) and electron donor/acceptor pairs (e.g.,carbazole/dinitrobenzene) demonstrate desirable properties inhydrophobic media, and, as such, can be successfully used in such anembodiment.

FIG. 5 provides an alternative synthesis pathway that uses anionic blockcopolymerization and subsequent functionalization of the 1,2polybutadiene vinyl groups. In this embodiment, the block copolymerscontain long PI segments and subsequent functionalization(n=7,000-15,000 and m, p=10-40 units.) R is a protecting group for thephenol moiety, e.g. t-butyl, acetyl, or t-butyldimethylsilyl. Themolecules on the left would be synthesized via anionic blockcopolymerization, and subsequent functionalization of the 1,2 PBsegments (to obtain the molecules on the right) would be achieved byradical addition of the corresponding thiols using AIBN. In thisembodiment, carbazole, dinitrobenzene, carboxylic acid, or tertiaryamine functionalities can be incorporated.

Although embodiments of the invention incorporating a telechelicsynthesis are described above, it should be understood that alternativesynthetic pathways may be used. For example, on an industrial scale, itmay cost effective to make the associative polymers of the instantinvention by condensation from prepolymers (P) with reactive ends andseparately produced associative clusters (A). In such an embodiment,when the associative precursors are present in excess, the resultingproduct will be a mixture that mainly contains the followingcombinations:

-   A-P;-   A-B-A;-   A-P-A-P-A; and-   small concentrations of still larger products.    The presence of species that contain the desired A-P-A motif within    them (such as A-P-A-P and A-P-A-P-A, etc.) also contribute to the    formation of desired supramolecules.

Although the above discussion has focused on mist control materials andtheir synthesis, the invention is also directed to methods ofsuppressing mist using the molecules set forth herein. Moreover, itshould be understood that, although an assumption is made throughoutthat the mist control agents of the instant invention will be used tosuppress misting in hydrocarbon fuels, one skilled in the art wouldrealize that this polymer design can be extended to other fluids (e.g.,aqueous media) to suppress misting and/or reduce drag.

As described above, the molecular design of the current inventionaddresses a number of deficiencies found in conventional polymeric mistcontrol additives and its use will allow for the effective control ofmist in a number of contexts:

-   The molecular design addresses the problem of flow induced    degradation. It has been demonstrated that 30 million MW polyolefins    chains introduced into fuel transportation pipes were consistently    degraded to chains of average MW 1.6 million featuring low    polydispersity. The current system is able to produce chains were    the upper limit of MW for degradation resistant chains will either    be above or within the 0.5-1 million MW range required for effective    mist control.-   The molecular design ensures that self-assembly results in    structures, which impart sufficient mist-control behavior to the    fuel. The molecules of the current invention are designed such that    both intramolecular interactions and loop formation are prohibited,    and associations are unsusceptible to chain collapse. Furthermore,    in a preferred embodiment, these molecules form a stoechiometric    blend of the building blocks, such that virtually all the polymer    chains will belong to linear pentamers independent of concentration    above a threshold value, which depends on the equilibrium constants    of the A+B and C+D associations. Using the result of Chao et al.,    which showed that linear polyisobutylene chains of molecular weight    6 million were very effective at reducing the flammability of sprays    of Jet-A at concentrations as low as 50 ppm, suggests that linear    aggregates of molecular weight 2.5 to 5 million will be effective    mist-control agents at concentrations as low as from 100-1000 ppm.    (See, e.g., Chao K. K., et al., AIChE J. 30 111-120 (1984), the    disclosure of which is incorporated herein by reference.)-   The molecular design accounts for the need to have aggregates form    again rapidly after breakup by the flow. The dynamics of aggregate    re-formation are governed by diffusion. Consider the A and B    endgroups of two model polymer chains (only one endgroup per chain)    of size 0.5 million MW at polymer concentrations of 250 ppm each, in    good solvent (this corresponds to the dilute regime.) Assuming that    the endgroups are ˜1 nm in size, i.e. on the order of the size of a    monomer, and that they ‘stick’ as soon as they come within that    distance of each other, it can be estimated that the time it takes    for 90% of the molecules of the instant invention to form dimers    would be of the order of only 1 sec (based on the diffusion    coefficient of the chains).

EXEMPLARY EMBODIMENTS

The person skilled in the art will recognize that additional embodimentsaccording to the invention are contemplated as being within the scope ofthe foregoing generic disclosure, and no disclaimer is in any wayintended by the foregoing, non-limiting examples.

Example 1: Development of a Theoretical Model

Herein, a theoretical underpinning is provided that describes howmolecules can be designed that overcome chain collapse by clusteringassociating groups at the ends of polymer chains. In particular, a modelis constructed that can predict for long linear chains endcapped withstrongly associating groups, the equilibrium partitioning of the polymerinto supramolecular chains and supramolecular loops of all sizes.

In the model, it is assumed that the A and B endgroups associate witheach other pair-wise with interaction energy εkT, but that neither the Anor the B endgroups self-associate. Under these assumptions, to modelthe equilibrium aggregation of telechelic polymers A----A and B----Binto supramolecular cyclic and linear chains of any length (see, schemeprovided in FIG. 6), the simpler case of association of telechelicpolymers A₁----A₂ and B₁----B₂l is first considered. In doing so, it canbe assumed that the end-groups A₁ and A₂, and likewise B₁ and B₂, aredistinguishable but of identical reactivity (as might be the case, forexample, if one atom of A₁ were a different isotope than thecorresponding atom of A₂). Using the lattice model provided, which isdescribed in greater detail below, it is possible to determine theequilibrium distribution of all the aggregates for a given energy ofassociation εkT. (Discussion of a comparable lattice model can be foundin Goldstein, R. E., Journal of Chemical Physics 1986, 84, (6),3367-3378, the disclosure of which is incorporated herein by reference.)

Consider a solution composed of N_(s) solvent molecules and N_(Atotal)and N_(Btotal) telechelic A₁----A₂ and B₁----B₂ chains, of respectivelength M_(A) and M_(B) elementary units (monomers). The solution volumeV is partitioned into lattice sites of volume a³′ where a³ is the volumeof a solvent molecule and also the volume of a monomer. The assumptioncan be made that there is no volume change upon mixing, so thatV=a³⁽N_(s)+N_(Atotal)M_(A)+N_(Btotal)M_(B))=Λa³, where Λ is the totalnumber of “sites.” In these calculations, the subscript s refers to thesolvent and the subscripts i or j refer to single-chain andsupramolecular components. Unless otherwise specified, sums Σ_(j) areover all polymer components in solution, i.e., the telechelic startingmaterials and all polymer aggregates. The volume fractions of solventand polymer component j are ϕ_(s)=N_(s/)L and ϕ_(j)=N_(j)M_(j/)L, whereM_(j) is the number of monomers of polymer component j. Letϕ=Σ_(j)M_(j)N_(j/)Λ=1−ϕ_(s) denote the total polymer volume fraction insolution. The center-of-mass and configurational entropy of the polymercomponents and solvent is:

$\begin{matrix}{S = {{k{\sum\limits_{j}{{ln\Omega}\left( {0,N_{j}}\  \right)}}} + {\Delta S_{mix}}}} & \left( {{EQ}.\mspace{14mu} 1} \right)\end{matrix}$

where Ω(0,N_(j)) is the number of possible configurations of N_(j)molecules of polymer component j, each of length M_(j), onto M_(j)N_(j)sites (referring to pure polymer before mixing), so that the sumaccounts for the entropy of all the polymer components before mixing.Here the notation of Hill has been retained for the entropy of a puresolution of N_(i) linear polymer chains of length M_(i) according to:

$\begin{matrix}{{\ln \; \Omega} = {\left( {0,N_{i}} \right) = {{{- N_{i}}\ln N_{i}} + N_{i} + {M_{i}N_{i}{\ln \left( {M_{i}N_{i}} \right)}} - {M_{i}N_{i}} + {{N_{i}\left( {M_{i} - 1} \right)}{\ln \left( \frac{c - 1}{M_{i}N_{i}} \right)}}}}} & \left( {{EQ}.\mspace{14mu} 2} \right)\end{matrix}$

where c is the coordination number, i.e., the number of sitesneighboring any given monomer where the next monomer on the chain may befound. (See, e.g., Hill, T. L., An Introduction to StatisticalThermodynamics, Dover Publications: 1986; pp. 402-404, the disclosure ofwhich is incorporated herein by reference.) The entropy of mixing of thesolvent and all polymer components, ΔS_(mix,) is approximated using theFlory-Huggins expression:

$\begin{matrix}{{\Delta S_{mix}} = {- {{k\left( {{N_{s}\ln \; \varphi_{s}} + {\sum\limits_{j}{N_{j}\ \ln \; \varphi_{j}}}} \right)}.}}} & \left( {{EQ}.\mspace{14mu} 3} \right)\end{matrix}$

Equation 1 does not account for the entropic cost of loop closure forsupramolecular cycles; that contribution will instead be absorbed intothe standard chemical potential of the cyclic components, as discussedlater. The entropic contribution to the mixture's free energy istherefore:

$\begin{matrix}{F_{S} = {{{{- T}\Delta S_{mix}} - {{kT}{\sum\limits_{j}{\ln \; {\Omega \left( {0,N_{j}} \right)}}}}} = {{k{T\left\lbrack {{N_{s}{\ln \left( \frac{N_{s}}{\Omega} \right)}} + {\sum\limits_{j}{N_{j}{\ln \left( \frac{N_{j}}{\Omega} \right)}}}} \right\rbrack}} + {{kT}{\sum\limits_{j}{N_{j}{\ln\left( M_{j}\  \right)}}}} - {{kT}{\sum\limits_{j}{\ln \; {{\Omega \left( {0,N_{j}} \right)}.}}}}}}} & \left( {{EQ}.\mspace{14mu} 4} \right)\end{matrix}$

Next, the contribution to the solution free energy due tosolvent-solvent, polymer-solvent, and polymer-polymer interactions isestimated by the random mixing approximation:

F _(int)=Ωδ[(1−ϕ)² h _(ss)+ϕ² h _(pp)+2ϕ(1−ϕ)h _(ps)]  (EQ. 5)

where δ is one-half the local coordination number, and h_(ij) are themicroscopic interaction energies of the polymer and solvent species. Thetotal free energy F of the solution is the sum of F_(S,) F_(int,) and ofcontributions from the internal free energy of solvent and polymercomponents:

$\begin{matrix}{F = {F_{int} + F_{S} + {N_{s}\mu_{s}^{0}} + {\sum\limits_{j}{N_{j}\mu_{j}^{0}}}}} & \left( {{EQ}.\mspace{14mu} 6} \right)\end{matrix}$

where μ_(i) ⁰ is the standard chemical potential of the single-chain orsupramolecular chain component i. Usingϕ=(M_(i)N_(i)+Σ_(j≠i)M_(j)N_(j))/Λ withΛ=N_(s)+M_(i)N_(i)+Σ_(j≠i)M_(j)N_(j), the contribution to the chemicalpotential of polymer component i due to interactions is:

$\begin{matrix}{{{\mu_{{int},i} = \frac{\partial F_{int}}{\partial N_{i}}}}_{N_{j \neq i}} = {{{- \omega}\; M_{i}\varphi_{s}^{2}} + {\omega_{pp}M_{i}}}} & \left( {{EQ}.\mspace{14mu} 7} \right)\end{matrix}$

where for convenience we have introduced ω_(mn)=δh_(mn) andω=ω_(pp)+ω_(ss)−2 ω_(ps). The entropic contribution to the chemicalpotential of polymer component i is:

$\begin{matrix}{{{\frac{\mu_{S,i}}{kT} = {\frac{1}{kT}\frac{\partial F_{S}}{\partial N_{i}}}}}_{N_{j \neq i}} = {{\ln \left( \frac{\varphi_{i}}{M_{i}} \right)} + 1 - \varphi_{i} - {M_{i}\left\lbrack {\varphi_{s} + {\sum\limits_{j \neq i}\frac{\varphi_{j}}{M_{j}}}} \right\rbrack} + {\ln M_{i}} - 1 - {M_{i}\left\lbrack {{\ln \left( {c - 1} \right)} - 1} \right\rbrack} - {\ln M_{i}} + {\ln \left( {c - 1} \right)}}} & \left( {{EQ}.\mspace{14mu} 8} \right)\end{matrix}$

or, after rearranging:

$\begin{matrix}{{{\frac{1}{kT}\frac{\partial F_{S}}{\partial N_{i}}}}_{N_{j \neq i}} = {{\ln \left( \frac{\varphi_{i}}{M_{i}} \right)} - {M_{i}\left\lbrack {\varphi_{s} + {\sum\limits_{j}\frac{\varphi_{j}}{M_{j}}}} \right\rbrack} + f_{i}}} & \left( {{EQ}.\mspace{14mu} 9} \right)\end{matrix}$

where f_(i)=ln(c−1)+M_(i)[1−ln(c−1)]. Differentiation of EQ. 6 andsubstitution of EQs. 7 and 9 give the following expression for thechemical potential of component i, valid for the single-chain buildingblocks and all aggregates:

$\begin{matrix}{\mu_{i} = {\left. \frac{\partial F}{\partial N_{i}} \right|_{N_{j \neq i}} = {\mu_{i}^{0} + {kT\left\{ {{\ln \left( \frac{\varphi_{i}}{M_{i}} \right)} - {M_{i}\left\lbrack {\varphi_{s} + {\sum\limits_{j}\frac{\varphi_{j}}{M_{j}}}} \right\rbrack} + f_{i}} \right\}} - {\omega \; M_{i}\varphi_{s}^{2}} + {\omega_{pp}{M_{i}.}}}}} & \left( {{EQ}.\mspace{14mu} 10} \right)\end{matrix}$

Consider a supramolecular component i made up of n_(i) A₁----A₂ andB₁----B₂, m_(i) B₁----B₂ building blocks: its size isM_(i)=n_(i)M_(A)+m_(i)M_(B). At the equilibrium partitioning of thetelechelic building blocks into aggregates of all size, its chemicalpotential satisfies the equilibrium condition:

μ_(i) n _(i)μ_(A) +m _(i)μ_(B)  (EQ. 11)

where μ_(A) and μ_(B) are the chemical potentials of building blocksA₁----A₂ and B₁----B₂, respectively. Substituting the expressions forμ_(i), μ_(A), and μ_(B) from EQ. 10 into EQ. 11 above, we obtain, afterrearrangement, the following mass-action relation for polymer componenti:

$\begin{matrix}{{\mu_{i}^{0} + {k{T\left\lbrack {{\ln \left( \frac{\varphi_{i}}{M_{i}} \right)} + f_{i}} \right\rbrack}}} = {{n_{i}\mu_{A}^{0}} + {m_{i}\mu_{B}^{0}} + {k{T\left\lbrack {{n_{i}{\ln \left( \frac{\varphi_{A}}{M_{A}} \right)}} + {m_{i}{\ln \left( \frac{\varphi_{B}}{M_{B}} \right)}} + {n_{i}f_{A}} + {m_{i}f_{B}}} \right\rbrack}}}} & \left( {{4.1}2} \right)\end{matrix}$

where ϕ_(A), ϕ_(B) are the volume fractions of the telechelic buildingblocks A₁----A₂ and B₁----B₂, respectively. Equation 12 above can berewritten as follows:

$\begin{matrix}{\left( \frac{\varphi_{i}}{{n_{i}M_{A}} + {m_{i}M_{B}}} \right) = {\left( \frac{\varphi_{A}}{M_{A}} \right)^{n_{i}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{i}}{\exp \left( \Gamma_{i} \right)}}} & \left( {{EQ}.\mspace{14mu} 13} \right) \\{{where}{\Gamma_{i} = {{\frac{1}{kT}\left( {{n_{i}\mu_{A}^{0}} + {m_{i}\mu_{B}^{0}} - \mu_{i}^{0}} \right)} + {\left( {n_{i} + m_{i} - 1} \right){{\ln \left( {c - 1} \right)}.}}}}} & \left( {{EQ}.\mspace{14mu} 14} \right)\end{matrix}$

The conservation equations are then:

$\begin{matrix}{\varphi_{Atotal} = {{\sum\limits_{j}{\varphi_{j}\left( \frac{n_{j}M_{A}}{{n_{j}M_{A}} + {m_{j}M_{B}}} \right)}} = {\sum\limits_{j}{n_{j}{M_{A}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{j}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{j}}{\exp \left( \Gamma_{j} \right)}}}}} & \left( {{EQ}.\mspace{14mu} 15} \right) \\{\varphi_{Btotal} = {{\sum\limits_{j}{\varphi_{j}\left( \frac{m_{j}M_{B}}{{n_{j}M_{A}} + {m_{j}M_{B}}} \right)}} = {\sum\limits_{j}{m_{j}{M_{B}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{j}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{j}}{{\exp \left( \Gamma_{j} \right)}.}}}}} & \;\end{matrix}$

In this model, terms arising from microscopic interactions, as well asterms arising from the center-of-mass and configurational entropy(except loop closure) of polymer components and solvent in solution havebeen carried out explicitly. On the other hand, terms arising from (i)the energy of association of the endgroups within a polymer aggregate,and (ii) the entropic cost of loop closure for cyclic supramolecularaggregates, are instead absorbed into the standard chemical potentialsof the aggregates. Expressions accounting for these effects are derivedseparately, below.

Theoretical Model: Entropic Cost of Loop Closure Calculation

The entropic cost of loop closure is determined by calculating theprobability of loop closure, as follows: For Gaussian linear chains of NKuhn monomers of length b, the probability density function for theend-to-end vector r is:³

$\begin{matrix}{{G_{Gaussian}\left( {r,N} \right)} = {\left( \frac{3}{2\; \pi \; {Nb}^{2}} \right)^{\frac{3}{2}}\mspace{14mu} \exp {\left\{ {- \frac{3r^{2}}{2Nb^{2}}} \right\}.}}} & \left( {{EQ}.\mspace{14mu} 16} \right)\end{matrix}$

The argument within the exponential is −3r²/(2Nb²)≅0 for∥r∥<<<r²>^(1/2), so the probability that the chain ends be within asmall distance x of each other, where x/b˜O(1), is:

$\begin{matrix}{G_{{cyc},{Gaussian}} = {{\left( \frac{3}{2\; \pi \; {Nb}^{2}} \right)^{\frac{3}{2}}{\int_{0}^{2\; \pi}{d\; \theta {\int_{0}^{\pi}{d\; {\theta sin}\; \theta {\int_{0}^{a}{{{dr} \cdot r^{2}}{\exp (0)}}}}}}}} = {{4\; {\pi \left( \frac{3}{2\; \pi \; N\; b^{2}} \right)}^{\frac{3}{2}}{\int_{0}^{a}{{{dr} \cdot r^{2}}{\exp (0)}}}} = {\left( \frac{6}{\pi \; N^{3}} \right)^{\frac{1}{2}}{\left( \frac{x}{b} \right)^{3}.}}}}} & \left( {{EQ}.\mspace{14mu} 17} \right)\end{matrix}$

For real chains, excluded volume interactions of the monomers at chainends reduce the probability density function G(r,N) by the factor

$\begin{matrix}{\frac{G_{real}\left( {r,N} \right)}{G_{Gaussian}\left( {r,N} \right)}\text{∼}\left( \frac{r}{\sqrt{\langle r^{2}\rangle}} \right)^{g}\mspace{14mu} {for}\mspace{14mu} \frac{r}{\sqrt{\langle r^{2}\rangle}}{\operatorname{<<}1}} & \left( {{EQ}.\mspace{14mu} 18} \right)\end{matrix}$

where the exponent g≅0.28,⁴ so that the probability of cyclizationbecomes

$\begin{matrix}{G_{{cyc},{real}} \approx {4{\pi \left( \frac{3}{2\pi Nb^{2}} \right)}^{\frac{3}{2}}\left( \frac{1}{{bN}^{v}} \right)^{g}{\int_{0}^{a}{{{dr} \cdot r^{2 + g}}{\exp (0)}\text{∼}N^{{{- 3}/2} - {gv}}}}}} & \left( {{EQ}.\mspace{14mu} 19} \right)\end{matrix}$

where the fractal exponent v is 0.588 in good solvent. The loop closureprobability thus scales as N^(−3/2) for Gaussian chains and N^(−1.66)for swollen chains. The entropic cost of loop closure is simplyΔS_(loop)=−klnG_(cyc).

In dilute or semi-dilute solutions, all chain segments smaller than thethermal blob g_(T)≈b⁶/v² (where v is the excluded volume parameter) havenearly Gaussian statistics because excluded volume interactions areweaker than the thermal energy. If, for a solution composed of anynumber of different (single and supramolecular) chains j of size M_(j),at total polymer volume fraction ϕ=Σ_(j)ϕ_(j), it is assumed that thepolymer chains are dilute enough to ignore polymer-polymer interactions,it is possible to use the following expression in the calculation of theentropic cost of loop closure ΔS_(loop)=−klnG_(cyc) for any cyclicaggregate q:

$\begin{matrix}{{G_{{cyc},q} \approx {\left( \frac{6}{\pi \; g_{T}^{3}} \right)^{\frac{1}{2}}\left( \frac{x}{b} \right)^{3}\left( \frac{M_{q}}{g_{T}} \right)^{- 1.66}}}.} & \left( {{EQ}.\mspace{14mu} 20} \right)\end{matrix}$

By doing so the assumption is being made that all chain segments largerthan gr are fully swollen.

Theoretical Model: Inventory of Polymer Components Calculation

As a starting point in classifying the polymer aggregates, they aregrouped together As a starting point in classifying the polymeraggregates, they are grouped together according to their size andtopology: all the components j that belong to any particular group ghave the same topology (either linear or cyclic) and have the same n_(i)(number of A₁----A₂ building blocks) and m_(i) (number of B₁----B₂building blocks). Consequently all members of a given group g have thesame size M_(i)=n_(i)M_(A)+m_(i)M_(B) and Γ_(j)=Γ_(g). The modelpredicts ϕ_(g) refer to the cumulative volume fraction of all thepolymer components that belong to group g. The equilibrium condition andthe conservation equations can be rewritten as:

$\begin{matrix}{\left( \frac{\varphi_{g}}{{n_{g}M_{g}} + {m_{g}M_{g}}} \right) = {{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}{\exp \left( \Gamma_{g} \right)}}} & \left( {{EQ}.\mspace{14mu} 21} \right) \\{{\varphi_{Atotal} = {\sum\limits_{g}{n_{g}M_{A}{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}\exp \left( \Gamma_{g} \right)}}}{\varphi_{Btotal} = {\sum\limits_{g}{m_{g}M_{B}{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}{\exp \left( \Gamma_{g} \right)}}}}} & \left( {{EQ}.\mspace{14mu} 22} \right)\end{matrix}$

where Ω_(g) refers to the number of distinct species in group g.

It is next necessary to determine how many components belong to eachgroup. For linear aggregates there are two possibilities: (i) ifn_(g)+m_(g) is even (i.e., n_(g)=m_(g)), then no sequence read from leftto right will be the same as a sequence read from right to left, so thenumber of ways to arrange the molecules is Ω_(g)=2^(n) ^(g) ^(+m) ^(g) ;(ii) if n_(g)+m_(g) is odd, then every sequence read from left to rightwill have a matching sequence read from right to left, so the number ofways to arrange the molecules is Ω_(g)=2^(n) ^(g) ^(+m) ^(g) ⁻¹.Supramolecular cycles can only be formed if n_(g)=m_(g). The number ofways to form such a loop is derived below; to a very good approximationit is Ω_(cyc,g)=2+(2^(2n) ^(g) ⁻¹−2)/n_(g).

Theoretical Model: Number of Ways to Form Loops Calculation

It is next necessary to determine the number of different loops that canbe formed by linking n A₁----A₂ and n B₁----B₂ telechelic chainsend-to-end via association of A and B endgroups, as shown in FIG. 7.Again, in this model the A₁ groups are being treated as distinguishablefrom A₂ groups, and likewise B₁ groups are distinguishable from B₂groups, but that the n A₁----A₂ molecules are indistinguishable, andlikewise are the n B₁----B₂ molecules. The question is equivalent to thecombinatorial problem of counting necklaces formed using beads ofdifferent colors, in which two necklaces are considered equivalent ifone can be rotated to give the other. The way to recognize theequivalence is to break up the loops into adjacent pairs of telechelics(with one A₁----A₂ and one B₁----B₂ molecule per pair), and map theloops into necklaces made up of n beads of 4 different “colors”corresponding to: A₁A₂B₁B₂, A₁A₂B₂B₁, A₂A₁B₁B₂, A₂A₁B₂B₁. For example,A₁A₂B₁B₂=black, A₁A₂B₂B₁=white, A₂A₁B₁B₂=blue, and A₂A₁B₂B₁=green.) Theformula for the number of different necklaces is:

$\begin{matrix}{{m(n)} = {\frac{1}{n}{\sum\limits_{d|n}\left\lbrack {{\phi (d)} \cdot 4^{n/d}} \right\rbrack}}} & \left( {{EQ}.\mspace{14mu} 23} \right)\end{matrix}$

where the sum is over all numbers d that divide n, and φ(d) is the Eulerphi function. (For a full discussion, see van Lint, J. H. and Wilson, R.M., In A Course in Combinatorics, Cambridge University Press: 2001; pp.522-525, the disclosure of which is incorporated herein by reference.)

In reality, the above formula over counts the number of ways to formpolymer loops by a factor of two, as explained in Chapter 4 of David, R.L. A., Ph.D. Dissertation, California Institute of Technology, Pasadena,Calif., 2008. The number of distinct loops s(n) that can be formed bylinking n A₁----A₂ and n B₁----B₂ telechelic chains end-to-end viaassociation of A and B endgroups is therefore:

$\begin{matrix}{{s(n)} = {\frac{1}{2n}{\sum\limits_{d|n}{\left\lbrack {{\phi (d)} \cdot 4^{n/d}} \right\rbrack.}}}} & \left( {{EQ}.\mspace{14mu} 24} \right)\end{matrix}$

Theoretical Model: Standard Chemical Potential of Polymer Aggregates

In EQs. 13 to 15 and 21 to 22, the expressions for the standard chemicalpotential of the aggregates μ_(j) ⁰ the contributions due to the energyof association of the endgroups and to the entropic cost of loop closurewere absorbed. The equation for the standard chemical potential of anypolymer component j within group g is therefore:

$\begin{matrix}{\mu_{g}^{0} = \left\{ {\begin{matrix}{{n_{g}\mu_{A}^{0}} + {m_{g}\mu_{B}^{0}} - {ɛ\; {{kT}\left( {n_{g} + m_{g}} \right)}} - {{kT}\; \ln \; G_{{cycl},g}}} & {{if}\mspace{14mu} {cyclic}} \\{{n_{g}\mu_{A}^{0}} + {m_{g}\mu_{B}^{0}} - {ɛ\; {{kT}\left( {n_{g} + m_{g} - 1} \right)}}} & {{if}\mspace{14mu} {linear}}\end{matrix},} \right.} & \left( {{EQ}.\mspace{14mu} 25} \right)\end{matrix}$

so that Γ_(g) in the equilibrium and conservation relationships (EQ. 21and 22) is:

$\begin{matrix}{\Gamma_{g} = \left\{ {\begin{matrix}{{ɛ\left( {n_{g} + m_{i}} \right)} + {\left( {n_{i} + m_{i} - 1} \right)\ln \; \left( {c - 1} \right)} + {\ln G_{{cycl},g}}} & {{if}\mspace{14mu} {cyclic}} \\{{ɛ\left( {n_{g} + m_{g} - 1} \right)} + {\left( {n_{g} + m_{g} - 1} \right)\ln \; \left( {c - 1} \right)}} & {{if}\mspace{14mu} {linear}}\end{matrix}.} \right.} & \left( {{EQ}.\mspace{14mu} 26} \right)\end{matrix}$

Theoretical Model: Distinguishable Versus Indistinguishable Endgroups

Consider the reversible association reactions of the individual chainsto form dimeric, linear supramolecules. Let ϕ_(A) and ϕ_(B) be thevolume fractions of the starting materials, and ϕ_(dimer) be the totalvolume fraction of product dimers. In the case that the chains that onlyhave associative groups at one end (A---- and B----), the equilibriumcondition (EQ. 13) is:

$\begin{matrix}{\left( \frac{\varphi_{dimer}}{M_{A} + M_{B}} \right) = {\left( \frac{\varphi_{A}}{M_{A}} \right)\left( \frac{\varphi_{B}}{M_{B}} \right){\exp (\Gamma)}}} & \left( {{EQ}.\mspace{14mu} 27} \right)\end{matrix}$

and in the case of telechelic chains having distinguishable ends, theequilibrium condition is:

$\begin{matrix}{\left( \frac{{1/4}\; \varphi_{dimer}}{M_{A} + M_{B}} \right) = {\left( \frac{\varphi_{A}}{M_{A}} \right)\left( \frac{\varphi_{B}}{M_{B}} \right){\exp (\Gamma)}}} & \left( {{EQ}.\mspace{14mu} 28} \right)\end{matrix}$

where M_(A) and M_(B) are the number of monomers in the startingmaterials, and Γ=e+ln(c−1) according to EQ. 26. In EQ. 28, ¼ ϕ_(dimer)is the volume fraction of each of the product dimers, so the differencebetween EQs. 27 and 28 simply reflects the difference in the number ofways to form dimers, i.e., Ω=4 for telechelic chains withdistinguishable ends, while Ω=1 for chains that are only functionalizedat one end. The correspondingly larger equilibrium fraction of dimers inthe case of telechelics with distinguishable ends can be intuitivelyunderstood to be a mere consequence of the increased contact probabilityof the endgroups to form the product: the rate of dissociation of dimersis equal in both cases, but the rate of association of reactants isexpected to be 4 times greater in the case of telechelic polymers

Next the equilibrium condition for the total volume fraction of productdimers in the case of indistinguishable ends (A----A and B----B) isdetermined. If the endgroups A, A₁, and A₂ have precisely the samereactivity, and likewise the endgroups B, B₁, and B₂, there cannot beany difference in the equilibrium partitioning of the molecules betweenthe two telechelic cases (distinguishable vs. indistinguishable ends),so that the equilibrium condition for the indistinguishable case is EQ.28, not EQ. 27. This argument can be generalized to conclude that theequilibrium among all the species in FIG. 7, where endgroups areindistinguishable, is mathematically the same as the solution which wasdeveloped for telechelics A₁----A₂ and B₁----B₂, where endgroups aredistinguishable. A less careful modeling of the association oftelechelic polymers A----A and B----B might miscalculate the cumulativeequilibrium volume fraction of polymer aggregates that fall within anygroup g by omitting the factor Ω_(g) in EQ. 21

Theoretical Model: Addition of End-Capping Chains

Addition of “end-capping chains” can be used to alter the relativepartitioning into linear vs. cyclic aggregates, and the model can beextended to capture that behavior. Consider the addition of N_(captotal)end-capping A---- chains, of length M_(cap) and total volume fractionϕ_(captotal). Let p_(j)=p_(g) refer to the number of end-capping A----chains in any polymer component j belonging to group g and let ϕ_(cap)be the equilibrium volume fraction of free A---- chains. The equilibriumcondition for any group g becomes:

$\begin{matrix}{\left( \frac{\varphi_{g}}{{n_{g}M_{A}} + {m_{g}M_{B}} + {p_{g}M_{cap}}} \right) = {{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}\left( \frac{\varphi_{cap}}{M_{cap}} \right)^{p_{g}}{\exp \left( \Gamma_{g} \right)}}} & \left( {{EQ}.\mspace{14mu} 29} \right) \\{\mspace{79mu} {with}} & \; \\{\Gamma_{g} = \left\{ \begin{matrix}{{ɛ\left( {n_{g} + m_{g}} \right)} + {\left( {n_{g} + m_{g} - 1} \right){\ln \left( {c - 1} \right)}} + {\ln \; G_{{cycl},g}}} & {{if}\mspace{14mu} {cyclic}} \\{\left( {n_{g} + m_{g} + p_{g} - 1} \right)\left\lbrack {ɛ + {\ln \left( {c - 1} \right)}} \right\rbrack} & {{if}\mspace{14mu} {linear}}\end{matrix} \right.} & \left( {{EQ}.\mspace{14mu} 30} \right)\end{matrix}$

where for endcapped aggregates Ω_(cap,g)=2^(n) ^(g) ^(+m) ^(g) /p_(g),and the expressions for Ω_(g) for cyclic and non-endcapped linearaggregates are given above. The conservation equations become:

$\begin{matrix}{\varphi_{Atotal} = {\sum\limits_{g}{n_{g}M_{A}{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}\left( \frac{\varphi_{cap}}{M_{cap}} \right)^{p_{g}}{\exp \left( \Gamma_{g} \right)}}}} & \left( {{EQ}.\mspace{14mu} 31} \right) \\{{\varphi_{Btotal} = {\sum\limits_{g}{m_{g}M_{B}{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}\left( \frac{\varphi_{cap}}{M_{cap}} \right)^{p_{g}}\exp \left( \Gamma_{g} \right)}}}\varphi_{ca{ptotal}} = {\sum\limits_{g}{P_{g}M_{cap}{\Omega_{g}\left( \frac{\varphi_{A}}{M_{A}} \right)}^{n_{g}}\left( \frac{\varphi_{B}}{M_{B}} \right)^{m_{g}}\left( \frac{\varphi_{cap}}{M_{cap}} \right)^{p_{g}}{{\exp \left( \Gamma_{g} \right)}.}}}} & \;\end{matrix}$

Example 2: Computational Example

Now that the theoretical model has been explained, it can be used totest whether linear chains possessing strongly associating endgroups canbe useful as mist-control additives to aviation fuel. In this section,computational results are given for a case of practical significancethat can be tested experimentally using anionically polymerizedtelechelics: monodisperse polyisoprene (PI) chains with stronglyassociating endgroups, in dilute solutions in Jet-A solvent.

In addition, it is shown computationally and/or experimentally usingmodified polybutadiene (PB) chains (shown in FIGS. 1b and 1c ) that indilute solution, i) self-associating polymer chains (e.g. moleculesshown in FIG. 1b ) collapse onto themselves, becoming less effectivemist-control agents than linear non-associating chains of the samemolecular weight, and ii) interpolymer complexes formed by blending e.g.proton-donating chains and proton accepting chains (shown in FIG. 1c ,right) only yield small improvements in mist-suppression compared to theunassociating parent chains (shown in FIG. 1c , left), even though theaggregates are 1-2 orders of magnitude larger in molecular weight.

Choice of Parameters

The model thus far has assumed that solvent molecules and polymerelementary units have the same size a³. In general, this cannot beexpected. If the volume of a solvent molecule (v_(s)) differs from thatof a monomer, it is important to determine the lattice size a, and howmany lattice sites M should be assigned to a polymer chain of aparticular molecular weight MW_(p).

In this example, a mixture of N_(s) solvent molecules and N_(p)monodisperse polymer chains of molecular weight MW_(p) is modeled. Topreserve the volume fraction and number densities of polymer and solventmolecules, it is required that N_(s)a³=N_(s)v_(s) andN_(p)Ma³=N_(p)(MW_(p)/MW_(o))v_(mon), where v_(mon) and MW_(o) are thevolume and molecular weight of a chemical monomer. These conditionsrequire that a³=v_(s) and M=(MW_(p)/MW_(o)) (v_(mon)/a³), meaning thatthe number and size of the elementary units into which the chains arebroken is determined by the solvent size v_(s). In this way, the freedomto map the polymer chain as desired is sacrificed to: for instance, useof EQs. 16 to 20 assumes that the polymer chain has been mapped into anequivalent freely jointed chain with M=MW_(p)/MW_(K) Kuhn monomers,where MW_(K) is the molecular weight of a Kuhn monomer.

An alternative is to model a solution of N_(p) monodisperse polymermolecules of given number density N_(p)/V and volume fraction ϕ_(p).This method lets the monomer size determine the lattice size andrenormalizes the number of solvent molecules, as follows. Preserving thepolymer volume fraction and number density requires thatN_(s,model)a³=N_(s,real)v_(s) and N_(p)Ma³=N_(p)(MW_(p)/MW_(o))v_(mon),where N_(s,real) and N_(s,model) are the number of real and modelsolvent molecules dissolving the polymer chains, and other parametersare the same as above. In this manner it is possible to map the polymerchains in any number of ways. The cost of that improvement is that thenumber density of the solvent molecules is not preserved, but since thatnumber does not appear in the equilibrium equations (see, e.g., EQs. 13& 14), it is not essential.

In the calculation that follows, the latter approach has been used andthe chain has been broken into M=MW_(p)/MW_(K) Kuhn monomers usinga³=v_(mon)MW_(K)/MW_(o)=v_(K), where v_(K) is the volume of a Kuhnmonomer, and N_(s,model)=N_(s,real)v_(s)/v_(K). More particularly, thepolymer is broken in Kuhn monomers of molecular weight MW_(K), and thelattice size set as a³=v_(K)=MW_(K)/(N_(A)ρ), where N_(A) is Avogadro'sconstant and ρ is the polymer density. For 1,4-PI polymer, ρ=0.83 g/cm³and MW_(K)=113 g/mol, giving a≈6.1 Å. (See, e.g., Rubinstein, M. andColby, R. H., In Polymer Physics, Oxford: 2003; p. 53, Table 2.1, thedisclosure of which is incorporated herein by reference.) To quantitatethe entropic cost of loop closure, numerical values are needed for theend-to-end distance x we require to close the loop and for the number ofmonomers in a thermal blob g_(T)≈b⁶/v². (For simplicity x/b=1 waschosen.) The excluded volume parameter v was estimated to be such thatv/b³≈0.10 for PI in Jet-A, giving g_(T)≈100. (This value is obtainedfrom viscosity measurements of 580 kg/mol polyisoprene chains, ofpolydispersity index 1.22, in Jet-A solvent, by rearrangement of thescaling relation ϕ*≈(b³/v)⁶ ^(v) ⁻³N¹⁻³ using swelling exponent v≈0.588and overlap concentration ϕ*≅0.0049. The number of Kuhn monomers N wasestimated using MW_(K)=113, and the overlap concentration was determinedaccording to the criterion [η]ϕ*≅1. See, e.g., Rubinstein, M. and Colby,R. H., Polymer Physics. Oxford: 2003, Equation 5.19, p. 176, thedisclosure of which is incorporated herein by reference.) Finally, it isassumed that the random walks of the chains on the lattice correspond toa coordination number of c=6.

Computations

The following procedure was used to calculate the volume fraction of allpolymer components (i.e., single-chain starting materials and aggregatesof all sizes) at equilibrium, for polymer solutions of A₁----A₂ andB₁----B₂ telechelics and A---- “end-cap” chains of specified molecularweights at specified initial concentrations ϕ_(Atotal), ϕ_(Btotal), andϕ_(captotal) (polymer components were grouped as shown in FIG. 7):

-   First, choose a number of groups T_(groups) to include in the    analysis (even though there is an infinite number of possible    polymer components, it is expected that above a certain size,    polymer aggregates will have negligible equilibrium volume fraction    and can therefore be ignored);-   Calculate n_(g), m_(g), M_(g), Ω_(g), G_(cyc,g) (if appropriate),    and Γ_(g) for polymer group g, for g=1 . . . T_(groups);-   Solve the three conservation equations, EQs. 31, for (ϕ_(A), ϕ_(B),    ϕ_(cap));-   Calculate ϕ_(g) for g=1 . . . T_(groups) using EQ. 29; and-   Repeat with a new value of T_(groups) twice that of the previous one    until changes in the calculated values of ϕ_(g) from one value of    T_(groups) to the next are negligible.

Results

To translate model results into terms relevant to experiment, theequilibrium distribution of aggregates is described in terms of theconcentration of the various size supramolecular species. In the contextof polymer-induced mist-suppression, all linear aggregates of a givenlength are equivalent, and all cyclic aggregates of a given length arelikewise equivalent. Therefore, the cumulative volume fractionsϕ_(linear)(MW) and ϕ_(loop)(MW) of all the linear species and of all thecyclic species of a given molecular weight MW will be used to evaluatethe impacts of the following parameters on rheological solutionproperties: binding energy, concentration and degree of polymerizationof the single-chain building blocks, and presence or absence of“end-capping” chains (FIGS. 8 to 14).

Case 1: Mixtures of A----A and B----B Chains Only

At the lowest level of complexity, solutions of telechelics of equalmolecular weights (MW_(A)=MW_(B)=MW_(p)) and equal initial volumefractions (ϕ_(Atotal)=ϕ_(Btotal)=ϕ_(total)/2) are considered. In thiscase, the problem is reduced to understanding the association behavioras a function of MW_(p), ϕ_(total), and the energy of interaction e.

Comparison of model results for MW_(p)=10⁶ g/mol (labeled 1000 k infigures) at total polymer volume fraction ϕ_(total) of 1400 ppm and 800ppm (FIGS. 8 and 9) demonstrates two important effects of total polymerconcentration. First, at fixed MW_(p) and ε, increasing concentrationresults in a higher fraction of the polymer becoming involved in largerlinear aggregates (compare first and second rows in FIGS. 8 and 9): thedecrease in ϕ_(linear) with increasing aggregate MW is not as sharp at1400 ppm, and the position of the peak in ϕ_(linear) vs. MW is shiftedto the right at 1400 ppm for each ε (most visibly for e=18, left columnof FIG. 9). Second, the relative partitioning of the polymer into linearrather than cyclic aggregates is insensitive to total polymerconcentration ϕ_(total).

Consider now the effect of the length of the individual chains (MW_(p)),by comparing results for 5×10⁵ chains at 1400 ppm (third row, FIGS. 8and 9) and 1×10⁶ g/mol chains at 800 ppm (second row). Theseconcentrations were chosen to correspond to one-fourth of the overlapconcentration of the single-chains, i.e., ϕ_(total)=¼f* based on therespective values of MW_(p). It can be observed that the shape of theϕ_(linear) vs. MW curves for both these systems is nearly identical, foreach value of ε investigated. On the other hand, the relative proportionof loops vs. linear chains is substantially higher for the shorterchains, due to the smaller entropic cost of cyclization for shorterloops.

Finally, the effect of the energy of association on the equilibriumdistributions is very pronounced (the columns of FIGS. 8 and 9 are inascending order by association energy). First, higher values of εstrongly increase the population of loops of all sizes, i.e., increasingε increases the relative fraction of loops compared to linearaggregates. Second, increasing ε greatly broadens the distribution ofϕ_(linear) vs. MW, decreasing the magnitude of the peak in thedistribution. At values of ε≤14, aggregates are few and the dominantcomponents are the telechelic building block themselves. At values ofε≥20, the dominant components are cycles of low MW, but the distributionof linear supramolecules is nearly flat, meaning that very largeaggregates have a significant cumulative volume fraction at equilibrium.Intermediate values of the energy of association, corresponding to16≤ε≤18, provide a balance of interactions strong enough to driveformation of large superchains and weak enough to accommodate asignificant population with unpaired ends (i.e., linear superchains).

Case 2: Mixtures of A----A, B----B, and A---- Chains

Important changes in the partitioning of the polymer occur asend-capping A---- chains are added to solutions of A----A and B----Btelechelics. A schematic of possible endcaps is provided in FIG. 10. Atthe lowest level of complexity, again, solutions of polymer additives ofequal molecular weight (MW_(A)=MW_(B)=MW_(cap)=MW_(p)) are assumed.Solution compositions that maintain equal number densities of A and Bendgroups, i.e., such that ϕ_(captotal)=2(ϕ_(Btotal)−ϕ_(Atotal)) areconsidered. Therefore, the total polymer fraction of A---- end-cappingchains, ϕ_(captotal), must be in the range from 0 to ⅔. DefineX=ϕ_(Atotal)/ϕ_(Btotal) as the ratio of telechelics A----A totelechelics B----B; that ratio decreases from 1 to 0 as the fraction ofA---- increases from 0 to ⅔.

Results obtained for solutions of MW_(p)=10⁶ g/mol at total volumefraction ϕ_(Atotal)+ϕ_(Btotal)+ϕ_(captotal)800 ppm (FIGS. 11 to 14)confirm that introducing end-caps favors the formation of linearspecies. At fixed ε, MW_(p), and ϕ_(total), the fraction of polymerinvolved in cycles decreases with increasing volume fraction of A----end-caps, as expected (see top row of FIGS. 11 to 14). This is true atall values of ε and occurs simply because the presence of A----components decreases the fraction of linear chains that can form loops.Note that the increase in the concentration of linear species uponaddition of A---- (offsetting the decrease in ϕ_(loop)) heavily favorsshort, rather than long aggregates: in fact, the population of very longlinear superchains is reduced by adding end-caps (decreasing X), andthis was also true at all values of ε (most visible in bottom row ofFIGS. 11 and 12). In other words, increasing the volume fractionϕ_(captotal) of end-capping chains causes a narrowing of thedistribution of linear aggregates, meaning that a higher fraction ofpolymer is involved in smaller linear supramolecules.

A striking qualitative difference between the binary (A----A+B----B) andthe ternary systems is in the behavior as ε→∞. In the absence ofend-capping A----, the ratio of linear to cyclic supramolecules vanishesas ε→∞ (FIG. 9). As the free energy penalty for leaving unpairedstickers diverges, no linear chains can survive in the absence ofend-caps. When end-caps are present, one is free to increase ε withoutextinction of linear species; instead as ε→∞ a limiting distribution isachieved (FIG. 14) in which doubly end-capped linear species and cyclicspecies equilibrate in a manner that can be quantitatively controlled bythe choice of the relative number of A---- single chains.

Example 3: Modeling Mist Control Additives

In this example, a set of parameter values are provided that can beincorporated into the model set forth in Examples 1 and 2 for which theequilibrium distribution of the polymer components is suitable formist-suppression applications. The model can be used to determinewhether or not the efficacy of ultra-long chains and the resistance toshear degradation of associative polymers can be combined, and, it willbe shown that the present model may be used to guide the selection ofchain lengths, association strengths, and mixture compositions that holdthe greatest promise.

Chao and coworkers reported that polyisobutylene chains of molecularweight˜5×10⁶ g/mol were satisfactory mist-suppressing agents atconcentrations as low as 50 ppm in kerosene. (See, e.g., Chao, K. K., etal., AICHE Journal 1984, 30, (1), 111-120, the disclosure of which isincorporated herein by reference.) Considering that cyclic polymerchains are expected only to be as effective as linear chains of halftheir size, the cumulative amount of linear species of MW≥5×10⁶ g/moland cycles of MW≥10×10⁶ g/mol should be 50 ppm or more. Given thataviation fuel is continuously circulated on the aircraft as a heattransfer fluid, the kinetics of equilibration may play a role, i.e., inpractice long linear aggregates may not achieve their equilibriumdistribution (as will be discussed below). If so, polymer designs andmixture compositions that maximize the equilibrium fraction of polymerinvolved in linear supramolecular aggregates in the 5-10×10⁶ g/mol rangemay lead to maximal mist suppression in practice.

Parameter Space

By restricting the level of complexity (choosing the A----A, B----B, andA---- building blocks to be of the same molecular weight MW_(p), and byrequiring that A and B endgroups have equal number densities insolution), the parameter space is reduced to 4 dimensions. Within theparameter space {MW_(p), ε, ϕ_(total), X}, the equilibrium partitioningof the polymer can be optimized for mist-control applications given theconstraints of the problem. Here the bounds that are imposed on MW_(p)and ϕ_(total) can be considered in the context of fuel additives.

First, the highest possible values of MW_(p) should be used, since atfixed ε and ϕ_(total) the total fraction of polymer trapped in loopsdecreases monotonically with increasing MW_(p). In reality theupper-bound of MW_(p) is limited by mechanical degradation of thepolymer. Unintentional chain scission of our telechelics would result inan excess of end-capping species that would greatly reduce the size ofsupramolecular chains that form. Literature on shear degradation showsthat flexible linear chains of less than a few million g/mol resistdegradation associated with flow through pumps and turbulent pipelineflow. Therefore, the limit MW_(p)=10⁶ g/mol was imposed as our upperbound and compared results with MW_(p)=0.5×10⁶ g/mol in order toquantitate sensitivity to changes in MW_(p).

Next, implementation of a polymer-based mist-control technology is notpossible unless changes in shear viscosity of the fuel due to polymeraddition are very small. Here the upper bound in total volume fractionof polymer is chosen to be one-fourth of the overlap concentration ofthe A----A and B----B, and A---- building blocks, recognizing thatsupramolecular chains formed by physical associations may reach orexceed their overlap concentration. So long as the very longsupramolecules remain below their particular c*, the shear viscosity ofthe solution is expected to remain within permissible bounds. For thetwo chain lengths selected above, this constraint imposes a maximumpolymer volume fraction of 800 ppm for MW_(p)=10⁶ g/mol and 1400 ppm forMW_(p)=5×10⁵ g/mol. To quantify the improvements in mist control arisingfrom increases in concentration, results for 10⁶ g/mol chains at both800 and 1400 ppm were compared.

With the above choices for MW_(p) and ϕ_(total), the problem may bereduced to two dimensions, ε and X, which were examined over theirphysically relevant ranges.

Implication for Mist-Control Applications

The criterion for optimal results with regard to mist-suppressingapplications corresponds to maximizing the equilibrium fraction ofpolymer involved in linear supramolecular aggregates in the 5-10×10⁶g/mol range. Two key features of the distributions that satisfy thisobjectives are:

-   (i) favorable partitioning of the polymer into linear rather than    cyclic aggregates, and-   (ii) a concentration vs. molecular weight curve for linear    aggregates that is narrowly distributed and centered around ˜5×10⁶    g/mol.

According to the above criteria, model results show that partitioning ofthe polymer into linear superchains is favored at higher values ofMW_(p) and ϕ_(total), as expected, but that ˜2-fold changes in MW_(p) orϕ_(total) about the ˜{10⁶ g/mol, 800 ppm} upper-boundary determined bythe problem constraints yield only small changes in the overall shapesof the ϕ_(linear), ϕ_(loop) distributions (compare first and secondrows, and first and third rows at fixed sin FIGS. 8 and 9). Effects ofthe energy of interaction were much more pronounced. For example, a≤15%change in ε from 14 to 16 yielded a dramatic change in the shape of thesize distribution of aggregates for all values of {MW_(p), ϕ_(total)}(FIG. 8).

The strong dependence of the size distribution of linear and cyclicaggregates on energy of interactions has important implications formist-control applications. For mixtures of A----A and B----B molecules,model predictions indicate that “good” results are only achieved in anarrow range of association energy, 16≤ε≤18. The following twocomplications immediately arise:

-   The preparation of telechelic chains of such length (˜10⁶ g/mol)    terminated with endgroups that all bind with a precise target    strength of interaction of such magnitude (>16 kT) poses a    tremendous synthetic challenge (see, full discussion below); and-   The strength of physical associations is strongly temperature    dependent and the operating temperature range of interest for    aviation fuel is very broad (−50 to +60° C.), so it is doubtful that    any system could be designed to maintain the binding energy of the    polymer endblocks within such a narrow range.

Addition of A---- end-capping chains to A----A+B----B mixtures solvesthe above problem, at least under equilibrium conditions. For instance,model results for 10⁶ g/mol chains at ϕ_(total)=800 ppm and X=0.5 (FIG.14) show that for any value of ε≥20, the equilibrium volume fraction oflinear superchains of 5×10⁶ g/mol is greater than 100 ppm and that of7×10⁶ g/mol superchains is greater than 50 ppm. This means that themist-suppression effectiveness of such a polymer solution underequilibrium conditions will be robust with respect to fluctuation in εdue to temperature variations or to variations in molecular structure ofthe endgroups. The model, therefore, indicates that it is possible tocreate polymers whose equilibrium distribution of aggregates providesthe requisite concentration of very long supramolecular chains over awide range of temperature. At this point, it is necessary to inquirewhether it is reasonable to expect that equilibrium partitioning of thepolymer will be found under conditions of practical import.

Time to Reach the Equilibrium Distribution

Earlier, values of {ε, X} were explored that allowed for theoptimization of the equilibrium distribution of polymer components. Indoing so, it is assumed that under conditions of practical importance,equilibrium is restored as fast it is disturbed. To determine how longit takes to reach the equilibrium partitioning of the polymer intoaggregates of all sizes it is necessary to start by considering thelifetime of a bond. The relaxation time τ₀˜ηb³/kT of a monomer insolution of shear viscosity ˜1 mPa·s is on the order of 10⁻¹⁰ sec, sothe lifetime of a donor-acceptor physical bond τ_(b)˜τ₀ exp(ε) is on theorder of 0.001-10 sec for ε=17-25. Therefore, even if it is assumed thatequilibrium could be reached with a mere 10³ bond breaking and bondforming events, for endgroups associating with energy 20-25 kT, thattime is on the order of 1-10⁴ s. Consider now that processes such asrecirculation of the fuel within an aircraft are expected to breakuppolymer aggregates down to individual components at intervals of a fewminutes during, for example, passage through pumps. Experimentationshows that in solutions of A----A plus B----B plus A----, polymer chainsreassociate into large superchains sufficiently rapidly to be used asmist-suppressing agents for aviation fuel.

From Telechelics to Heterotelechelics

Using the model it was possible to determine that ternary mixtures ofA----A, B----B, and A---- chains in dilute solutions deserve the effortrequired to synthesize the polymer. Unfortunately, they suffer to someextent from the same problems as mixtures of A----A and B----B chainsonly (whose binary mixtures do not appear to be viable candidates formist-suppressing applications):

-   (i) To provide 50 ppm of superchains >5×10⁶ g/mol, the overall    polymer concentration must be several hundred ppm because a lot of    polymer is “lost” in useless small cycles and small linear    aggregates, and-   (ii) The reassembly of superchains takes time, and the longer the    superchain the longer it takes for its population to build up to its    equilibrium value.

The above insight suggests that a better alternative involves the designof a polymer system for which loops are prohibited and associationswould result in the systematic formation of well-defined linear chains.An example of such a design is shown in FIG. 3. It involves two sets ofspecific interactions, such that A endgroups interact only with Bendgroups, and C endgroups likewise only with D endgroups. The moleculesin FIG. 3 are designed such that for a stoichiometric blend of thebuilding blocks and at high enough binding affinity of the A+B and C+Dassociations, nearly all the polymer chains should assemble intopentamers (in 4 bond-forming events only) even at arbitrarily lowpolymer volume fraction ϕ_(total). As a result, satisfactory mistsuppression could be achieved with <100 ppm of A----A, B----C, and D----chains of size MW=10⁶ g/mol.

Nature of the Endgroups

Now, it is important to model what chemical moieties might be used atthe chain ends to generate association energies in the 17-25 kT range.For this exercise, let A and B refer to the small moleculescorresponding to these endgroups. In a first inquiry, it is necessary todetermine what equilibrium constant of association K_(ass) the aboveassociation energies correspond to. For the association reaction of thefree-endgroups A+B→AB, our model predictions are:

$\begin{matrix}{{\frac{\varphi_{AB}}{M_{AB}} = {{\left( \frac{\varphi_{A}}{M_{A}} \right)\left( \frac{\varphi_{B}}{M_{B}} \right){\exp \left\lbrack {{- \frac{1}{kT}}\left( {\mu_{AB}^{0} - \mu_{A}^{0} - \mu_{B}^{0}} \right)} \right\rbrack}} = {\exp (ɛ)}}}.} & \left( {{EQ}.\mspace{14mu} 32} \right)\end{matrix}$

For small molecules in dilute solution, this expression is consistentwith the equilibrium condition for ideal solutions (Raoult's law),x_(AB)/x_(A)x_(B)=exp[−(μ_(AB) ⁰−μ_(A) ⁰−μ_(B) ⁰)], where x is molefraction. It follows from EQ. 32 that K_(ass)≡C_(AB)/C_(A)C_(B)=υ_(s)exp(ε), where C is molar concentration and υ_(s) is the molar volume ofthe solvent. Thus, achieving binding energies of the endgroups in therange of ε=17-25 corresponds to association constants K_(ass) on theorder of 10⁷ to 10¹⁰ M⁻¹.

Although interacting chemical structures of binding constants up to 10⁶M⁻¹ are known (FIG. 15, and described in Binder, W. H. et al.,Macromolecules 2004, 37, (5), 1749-1759; and Kolomiets, E. et al.,Macromolecules 2006, 39, (3), 1173-1181.), the disclosures of which areincorporated herein by reference). However, in the past the syntheticchallenge of preparing telechelic polymer chains of size 10⁶ g/mol withwell-defined endgroups of binding constants ˜10⁷ to 10¹⁰ M⁻¹ has proventoo difficult. In addition to the challenge of finding a suitabledonor-acceptor pair, synthesis of telechelics becomes increasingly moredifficult with increasing size. Furthermore, the endgroups (which wouldbe present at <ppb levels in dilute polymer solutions) can be poisonedby even minute amounts of acids, bases, metals etc., present in thesolvent, thereby rendering the polymer ineffective.

The current invention addresses these synthetic problems by using shortpolymer endblocks featuring an arbitrary number of donor-acceptor typefunctional groups. For example, synthesis of 10⁶ g/mol polymer chainsendcapped with 1,2-PB endblocks of a few thousands g/mol enables thepreparation of associating polymer of tunable binding affinities bypost-polymerization functionalization of the 1,2-PB. This strategyprovides more flexibility in the choice of binding energy and alsofacilitates fast and effective optimization of material properties viarapid adjustments in the number and the identity of the functionalside-groups.

Example 3: Rheological Study

The solution rheology of a pair of proton-donating/accepting end-to-endassociating polymers (100KA10 and 100KN50) in a widely used aviationfuel Jet-A was studied at 40° C. under steady-state shear field. 100KA10denotes a tri-block copolymer poly(acrylicacid)-block-poly(cyclooctene)-block-poly(acrylic acid) (averagemolecular weight=10⁵ g mole⁻¹), in which the total number of acrylicacid units is 10; likewise, 100KN50 denotes a tri-block copolymerpoly(2-(dimethylamino)ethylmethacrylate)-block-poly(cyclooctene)-block-poly(2-(dimethylamino)ethylmethacrylate) (average molecular weight=10⁵ g mole⁻¹), in which thetotal number of 2-(dimethylamino)ethyl methacrylate units is 50. Bothpolymers were prepared by the combination of ring-open metathesispolymerization (ROMP) of cyclooctene and atom-transfer radicalpolymerization of acrylate monomers. Solutions of 100KA10 and 100KN50 inJet-A were prepared respectively at a weight concentration of 1.4 wt %,which is close to the overlap concentration. A 1:1 mixture of the twosolutions was then prepared to study the effect of complementaryend-to-end association. FIG. 16 shows a plot of the relationship betweenshear viscosity and shear rate for all the three solutions. At low shearrate regime, the solution of complementary pair exhibits a significantlyhigher shear viscosity than solutions of the two components at the sameconcentration. It is well known that the shear viscosity of polymersolutions is proportional to the half-power of apparent molecular weightof polymer chains. The observed increase in viscosity reflects theformation of long-chain supramolecular aggregates due to end-to-endcomplementary association. At high shear regime, the difference inviscosity between the blend and components vanishes. Although not to bebound by theory, it is possible that this is because the shear forceoutstrips the strength of end-to-end complementary association.

Example 4: Preparation of Poly (Acrylic acid)-block-Poly(Cyclooctene)-block-Poly (Acrylic acid) Triblock Proton-donatingAssociating Polymer (100KA10)

A macro initiator for ATRP of tert-butyl acrylate (the precursor ofassociating group acrylic acid) was prepared as follows. A 100 mlround-bottom Schlenk reactor equipped with a magnetic stir bar wascharged with 2.0 g (0.0172 moles) of cis-cyclooctene, 0.247 g (0.00069moles) of cis-2-butene-1,4-diyl bis(2-bromopropanoate), and 17 ml ofanhydrous dichloromethane. The content was degassed by freeze-pump-thawcycles. The polymerization was initiated by the addition of a solutionof 2.93 mg (0.0034 mmoles) second-generation Grubbs catalyst in 1.0 mlof degassed anhydrous dichloromethane. After 20 hours at 40° C., theresulting solution was precipitated drop-wise into stirring methanol(200 ml) at 0° C. The resulting polymer was then redissolved in 10 ml oftetrahydrofuran (THF) and reprecipitated into cold methanol twice. Theisolated polymer was dried in vacuum to remove any traces of solvent.Proton NMR (300 MHz, CDCl₃, room temperature) showed signals ofATRP-capable end groups 2-bromopropanoate at δ 4.36 (quartet, assignedto —CO₂CH—Br)) and δ 4.65 (multiplet, assigned to —CH₂—CO₂). Gelpermeation chromatography (THF, 30° C.) indicated that the polymer has anumber-average molecular weight of 12800 g mole⁻¹ and a PDI of 1.87.

The aforementioned macro initiator was used to initiate ATRP oftert-butyl acrylate, and the resulting triblock copolymer was then usedas macro chain transfer agent in the preparation of 100KA10, which isdescribed in detail below. A 10 ml tubular Schlenk reaction equippedwith a magnetic stir bar was charged with 0.25 g (0.0195 mmoles) of theaforementioned macro initiator, 0.20 g (1.56 mmoles) of tert-butylacrylate, 7 mg (0.039mmoles) of chelating agentpentamethyldiethylenetriamine (PMDETA), and 1 ml of anhydrous THF. Themixture was degassed by three freeze-pump-thaw cycles. 3.4 mg (0.0195mmoles) ATRP catalyst copper(I) bromide (CuBr) was loaded into thereactor under protection of argon flow when the mixture remained frozen.The reaction mass was then brought to room temperature and stirred for20 minutes to allow CuBr to complex with PMDETA and dissolve into theliquid phase. The reactor was placed in an oil bath at 66° C. toinitiate ATRP of tert-butyl acrylate. After 5 hours, the reaction wasterminated by exposing the reaction mass to air and dilution with 10 mlof THF. The resulting solution was passed through an active alumina(basic type) plug to remove metal catalyst. The filtrate was dried undervacuum at 40° C. overnight to remove unreacted tert-butyl acrylatemonomer. Proton NMR (300 MHz, CDCl₃, room temperature) showed signals ofpoly (tert-butyl acrylate) block at δ 2.22 (broad peak, assigned to—CHCO₂C) and δ 1.42 (singlet, assigned to tert-butyl group). GPC showedthat the polymer has a number-average molecular weight of 14100 g mole⁻¹and a PDI of 1.85. The average number of tert-butyl acrylate units perchain was estimated to be 10 based on the difference in molecular weightbetween the macro initiator and the resulting triblock.

To prepare the final polymer, the triblock above was used as macro chaintransfer agent in ROMP of cis-cyclooctene. A 100 ml round-bottom Schlenkreactor equipped with a magnetic stir bar was charged with 1.23 g(0.0106 moles) of cis-cyclooctene, 0.109 g (0.0106 mmoles) of the abovetriblock, and 11 ml of anhydrous dichloromethane. The content wasdegassed by three freeze-pump-thaw cycles. The polymerization wasinitiated by the addition of a solution of 0.94 mg (0.0011 mmoles)second-generation Grubbs catalyst in 1.0 ml of degassed anhydrousdichloromethane. After 20 hours at 40° C., the resulting solution wasprecipitated drop-wise into stirring methanol (200 ml) at 0° C. Theresulting polymer was then redissolved in 10 ml of tetrahydrofuran (THF)and reprecipitated into cold methanol twice. The isolated polymer wasdried in vacuum to remove any traces of solvent. Proton NMR (300 MHz,CDCl₃, room temperature) showed signals of poly (tert-butyl acrylate)block at δ 1.42 (singlet, assigned to tert-butyl group). Gel permeationchromatography (THF, 30° C.) indicated that the polymer has aweight-average molecular weight of 104000 g mole⁻¹ and a PDI of 1.73. Inorder to remove tert-butyl groups, 0.35 g of the resulting polymer wasmixed with 0.5 ml of trifluoroacetic acid, 10 ml of dichloromethane, andtrace butylhydroxytoluene (BHT) in a 50 ml round-bottom flask andstirred at room temperature overnight. The polymer was isolated by threetimes of reprecipitation in cold methanol. Proton NMR (300 MHz, CDCl₃,room temperature) showed that the signal of tert-butyl groupsdisappeared after deprotection. The proton-donating polymer 100KA10 wasthus obtained. Solubility test of 100KA10 in aviation fuel Jet-A attemperatures ranging from 25 to 50° C. showed that clear solutions couldbe achieved at concentrations as high as 1.5 wt %.

Example 5: Preparation of Poly (2-(dimethylamino)ethylmethacrylate)-block-Poly (cyclooctene)-block-Poly (2-(dimethylamino)ethyl methacrylate) Triblock Proton-accepting Associating Polymer,(100KN50)

The macro-CTA approach for 100KA10 was also utilized to prepare 100KN50.Same macro ATRP initiator (M_(n)=12800, PDI=1.87) was used to initiateATRP of 2-(dimethylamino)ethyl methacrylate for preparing macro CTAbearing proton-accepting groups. A 10 ml tubular Schlenk reactionequipped with a magnetic stir bar was charged with 0.20 g (0.0156mmoles) of the said macro initiator, 0.20 g (1.25 mmoles) of2-(dimethylamino)ethyl methacrylate, 7 mg (0.0312 mmoles) of chelatingagent 1,1,4,7,10, 10-hexamethyltriethylenetetramine (HMTETA), and 1 mlof anhydrous THF. The mixture was degassed by three freeze-pump-thawcycles. 3.4 mg (0.0195 mmoles) ATRP catalyst copper(I) bromide (CuBr)and 0.7 mg (0.062 mmoles) of copper(II) bromide (CuBr₂) were loaded intothe reactor under protection of argon flow when the mixture remainedfrozen. The reactor mass was then brought to room temperature andstirred for 20 minutes to allow CuBr/CuBr₂ to complex with HMTETA andthen dissolve into the liquid phase. The reactor was placed in an oilbath at 40° C. to initiate ATRP of 2-(dimethylamino)ethyl methacrylate.After 22 hours, the reaction was terminated by exposing the reactionmass to air and dilution with 10 ml of THF. The resulting solution waspassed through an active alumina (basic type) plug to remove metalcatalyst. The filtrate was dried under vacuum at 40° C. overnight toremove unreacted 2-(dimethylamino)ethyl methacrylate monomer. Proton NMR(300 MHz, CDCl₃, room temperature) showed signals of poly(2-(dimethylamino)ethyl methacrylate) blocks at δ 2.27 (singlet,assigned to —N(CH₃)₂), δ 2.55 (singlet, assigned to —CH₂—N), and δ 4.05(singlet, assigned to —CO₂—CH₂). Average number of2-(dimethylamino)ethyl methacrylate units per chain was estimated to be50 based on proton NMR results.

Preparation of proton-accepting triblock copolymer is described asfollows. A 100 ml round-bottom Schlenk reactor equipped with a magneticstir bar was charged with 2.91 g (0.025 moles) of cis-cyclooctene, 0.16g (0.025 mmoles) of the above macro CTA, and 25 ml of anhydrousdichloromethane. The content was degassed by three freeze-pump-thawcycle. The polymerization was initiated by the addition of a solution of1.3 mg (0.0015 mmoles) second-generation Grubbs catalyst in 1.0 ml ofdegassed anhydrous dichloromethane. After 20 hours at 40° C., theresulting solution was precipitated drop-wise into stirring methanol(200 ml) at 0° C. The resulting polymer was then redissolved in 10 ml oftetrahydrofuran (THF) and reprecipitated into cold methanol twice. Theisolated polymer was dried in vacuum to remove any traces of solvent.Signals of poly (2-(dimethylamino)ethyl methacrylate) block were allpresent in proton NMR of the resulting polymer. Gel permeationchromatography (THF, 30° C.) indicated that the polymer has aweight-average molecular weight of 102000 g mole⁻¹ and a PDI of 1.54.The proton-donating polymer 100KN50 was thus obtained. Solubility testof 100KN50 in aviation fuel Jet-A at temperatures ranging from 25 to 50°C. showed that clear solutions could be achieved at concentrations ashigh as 1.5 wt %.

CONCLUSION

Mist-Controlled Kerosene is well-established as a very promisingtechnology for improving the safety and security of our aviationsystems, but nonviable to date for lack of a proper material. Thecurrent invention provides a molecular design/architecture whichovercomes limitations of previous materials and which from the onsetaddresses key issues such as flow degradation and aggregation dynamics.Accordingly, in accordance with the current invention mist controlagents can be produced that improve the fire safety of aviation fuel andother substances, by reduce misting of the fuel, thereby delaying andreducing the intensity of fire immediately following the crash of a jetaircraft. (See, e.g., Fuel Safety Research”, Workshop Proceedings, FAA,29 Oct.-1 Nov. 1985, Alexandria, Va.; and “Research on Antimisting Fuelfor Suppression of Post-Crash Fires,” Virendra Sarohia, et al., AIAApaper 86-0573, the disclosures of which are incorporated herein byreference.) The proposed polymer additive may also serve to reduce thecost of transporting fuel and increase the throughput of fueltransported in existing pipes (e.g. reducing plane turnaround time owingto faster fueling.) This additional benefit is due to thewell-established decrease in the frictional losses under turbulent flowthat results from the addition of part-per-million (ppm) levels of longchain polymer to a fuel. (See, e.g., Lumley J. L., Annu. Rev. FluidMech. 1, 367-384 (1969); and Gyr A., H. W. Bewersdorff, Drag Reductionof Turbulent Flows by Additives; Kluwer (1995), the disclosures of whichare incorporated herein by reference.)

DOCTRINE OF EQUIVALENTS

Those skilled in the art will appreciate that the foregoing examples anddescriptions of various preferred embodiments of the present inventionare merely illustrative of the invention as a whole, and that variationsin the steps and various components of the present invention may be madewithin the spirit and scope of the invention. For example, it will beclear to one skilled in the art that additional components, alternativeconfigurations or other synthesis schemes would not affect the improvedproperties of the mist control agents of the current invention, norrender the mist control agents unsuitable for their intended purpose.Accordingly, the present invention is not limited to the specificembodiments described herein but, rather, is defined by the scope of theappended claims.

1. A material comprising: a mixture of polymers each having: at leastone non-associative chain having a weight average molar mass comprisedbetween 100 to 1000 kg/mol, said non-associative chain being selected toconfer solubility to the polymer in a particular medium such that themedium and the material remain as a homogeneous single phase duringstorage and transport; a plurality of associative groups disposed on atleast one end of each of said non-associative chains, wherein theassociative groups are selected such that the polymer does notself-associate, and wherein a plurality of associative groups separateeach of said non-associative chains; wherein the associative groups areselected from the group of hydrogen bond donor/acceptor pairs, chargetransfer donor/acceptor pairs and ionic bond donor/acceptor pairs;wherein the associative groups of said polymers reversibly associatewith an association strength less than that of a covalent bond to formsupramolecules.
 2. The material of claim 1, wherein the mixture ofpolymers comprises at least two different complementary polymers each ofsaid polymers comprising: at least one non-associative chain, and aplurality of associative groups dispose at either end of each of saidnon-associative chains; and wherein the associative groups of the atleast two different complementary polymers reversibly associate with anassociation strength less than that of a covalent bond to formsupramolecules.
 3. The material of claim 1, wherein the plurality ofassociative groups are disposed in close proximity to form associativeclusters at either end of said non-associative chain.
 4. The material ofclaim 3, wherein each of said clusters comprises at least twentyassociating groups.
 5. The material of claim 3, wherein the clusters areformed using a mode selected from the group consisting of comonomers,dendrimers, nanoparticies and specially designed chemical units thatconfer polyvalent association.
 6. (canceled)
 7. The material of claim 1,wherein the medium is fuel and the non-associative chains confersolubility in fuel over a wide temperature range −40° C. to +60° C. 8.The material of claim 7, wherein the solubility in fuel is conferred tothe middle chains by unsaturated hydrocarbon groups attached thereto. 9.The material of claim 2, wherein the at least different complementarypolymers form a donor/acceptor pair.
 10. (canceled)
 11. The material ofclaim 10, wherein the associative groups include a carboxylicacid/tertiary amine acid/base pair.
 12. The material of claim 11,wherein the tertiary amine is selected from the group consisting ofdialkylamino groups connected via a linker that includes methylenecarbons adjacent to the amine, including, (dimethylamino) alkyl groups,(diethylamino) alkyl groups, (methylethylamino) alkyl groups, andpyridyl groups.
 13. The material of claim 1, wherein the associativegroups include a hydrogen bond pair selected from the group consistingof hydroxyphenyl/pyridyl pairs, and2,4-Bis(propylamido)pyridine/Hydrouracil pair.
 14. The material of claim1, wherein the associative groups include a charge transfer pair, andwherein the electron donor is selected from the group consisting ofcarbazole derivatives and N,N-dimethylanilino derivatives, and whereinthe electron acceptor is selected from the group consisting ofdinitrophenyl group, 3,6-dinitro-carbazolyl group, buckminsterfullerene(C₆₀), 4-bromo-1-naphthyl group, dansyl group, anthracene, pyrene,2,3,5,6-tetrafluorophenyl group, and cyanophenyl group.
 15. The materialof claim 1, wherein the long chains are homopolymers or copolymers ofdifferent monomers.
 16. The material of claim 15, wherein the monomersare selected from the group of isoprene, butadiene, ethylene, propylene,butene, norbornene derivatives, cyclobutene, cyclopentene, cyclooctene,cyclooctadiene, and trans,trans,cis-1,5,9-cyclododecatriene.
 17. Thematerial of claim 1, wherein the association of the complementarypolymers is sufficiently strong to provide association of the polymersinto supramolecules at temperatures up to 60° C.
 18. The material ofclaim 2, wherein the mixture comprises two complementary polymers, andwherein the associative groups at each end of each of the complementarypolymers is different such that each end of each of the complementarypolymers is designed to associate with only one end of the othercomplementary polymer.
 19. The material of claim 18, wherein theassociative groups at the first end of both complementary polymersassociate by an acid/base interaction, and wherein the associativegroups at the second end of both complementary polymers associate by anelectron donor/acceptor interaction.
 20. The material of claim 19,wherein the complementary polymers selectively associate to formsupramolecules that are linear pentamers.
 21. The material of claim 1,wherein the polymers are provided in the mixture in a concentration offrom 100 to 1000 ppm. 22-36. (canceled)
 37. The material of claim 1,wherein the at least one non-associative chain comprises acrylate repeatunits.
 38. The material of claim 37, wherein each acrylate repeat unitis a methacrylate or a tert-butyl acrylate.
 39. The material of claim 1,wherein the supramolecules can reversibly dissociate without chemicaldegradation.
 40. A material comprising: a mixture of polymers eachhaving: at least one non-associative chain selected to confer solubilityto the polymer in a particular medium such that the medium and thematerial remain as a homogeneous single phase during storage andtransport; wherein the at least one non-associative chain comprisesacrylate repeat units; a plurality of associative groups disposed on atleast one end of each of said non-associative chains, wherein theassociative groups are selected such that the polymer does notself-associate, and wherein a plurality of associative groups separateeach of said non-associative chains; wherein the associative groups areselected from the group of hydrogen bond donor/acceptor pairs, chargetransfer donor/acceptor pairs and ionic bond donor/acceptor pairs;wherein the associative groups of said polymers reversibly associatewith an association strength less than that of a covalent bond to formsupramolecules.
 41. The material of claim 40, wherein the at least onenon-associative chain having a weight average molar mass comprisedbetween 100 to 1000 kg/mol.